diff --git a/src/sage/combinat/designs/bibd.py b/src/sage/combinat/designs/bibd.py
index 238254209e1..4bf9450280f 100644
--- a/src/sage/combinat/designs/bibd.py
+++ b/src/sage/combinat/designs/bibd.py
@@ -141,9 +141,9 @@ def balanced_incomplete_block_design(v,k,existence=False,use_LJCR=False):
     For `k > 5` there are currently very few constructions::
 
         sage: [v for v in xrange(150) if designs.balanced_incomplete_block_design(v,6,existence=True) is True]
-        [1, 6, 31, 91]
+        [1, 6, 31, 81, 91, 121]
         sage: [v for v in xrange(150) if designs.balanced_incomplete_block_design(v,6,existence=True) is Unknown]
-        [51, 61, 66, 76, 81, 96, 106, 111, 121, 126, 136, 141]
+        [51, 61, 66, 76, 96, 106, 111, 126, 136, 141]
 
     But we know some inexistence results::
 
@@ -466,6 +466,8 @@ def BIBD_from_TD(v,k,existence=False):
 
     return BIBD
 
+
+
 def BIBD_from_difference_family(G, D, check=True):
     r"""
     Return the BIBD associated to the difference family ``D`` on the group ``G``.
@@ -484,7 +486,7 @@ def BIBD_from_difference_family(G, D, check=True):
 
     - ``G`` - a finite additive Abelian group
 
-    - ``D`` - a difference family on ``G``.
+    - ``D`` - a difference family on ``G`` (short blocks are allowed).
 
     - ``check`` - whether or not we check the output (default: ``True``)
 
@@ -519,14 +521,18 @@ def BIBD_from_difference_family(G, D, check=True):
          [19, 20, 2, 12, 14],
          [20, 0, 3, 13, 15]]
     """
-    r = {e:i for i,e in enumerate(G)}
-    bibd = [[r[G(x)+g] for x in d] for d in D for g in r]
+    from difference_family import orbit_representatives, block_stabilizer
+    bibd = []
+    p_to_i = {g:i for i,g in enumerate(G)}
+    for b in D:
+        b = map(G,b)
+        for g in orbit_representatives(block_stabilizer(G,b),G):
+            bibd.append([p_to_i[G(i)+g] for i in b])
     if check:
         assert _check_pbd(bibd, G.cardinality(), [len(D[0])])
     return bibd
 
 
-
 ################
 # (v,4,1)-BIBD #
 ################
diff --git a/src/sage/combinat/designs/database.py b/src/sage/combinat/designs/database.py
index 00be99bd24b..0cc9294b2bb 100644
--- a/src/sage/combinat/designs/database.py
+++ b/src/sage/combinat/designs/database.py
@@ -3344,254 +3344,297 @@ def _test_Vmt_constructions():
     """
     pass
 
-def CDF_21_5_1():
-    r"""
-    A cyclic `(21,5,1)`-difference family.
-
-    From the examples VI.16.16 of [DesignHandbook]_.
-
-    EXAMPLES::
-
-        sage: from sage.combinat.designs.database import CDF_21_5_1
-        sage: from sage.combinat.designs.difference_family import is_difference_family
-        sage: G, D = CDF_21_5_1()
-        sage: is_difference_family(G,D,21,5,1)
-        True
-
-    The difference family is available from the constructor::
-
-        sage: _ = designs.difference_family(21,5,1)
-    """
-    D = [[0,1,4,14,16]]
-    from sage.rings.finite_rings.integer_mod_ring import Zmod
-    return Zmod(21), D
-
-def ADF_5x5_4_1():
-    r"""
-    An Abelian difference family on `(\ZZ / 5 \ZZ)^2`.
-
-    This is the example 7.26 of [Stinson2004]_.
-
-    EXAMPLES::
-
-        sage: from sage.combinat.designs.database import ADF_5x5_4_1
-        sage: from sage.combinat.designs.difference_family import is_difference_family
-        sage: G,D = ADF_5x5_4_1()
-        sage: is_difference_family(G,D,25,4,1)
-        True
-    """
-    D = [[(0,0),(0,1),(1,0),(2,2)],[(0,0),(0,2),(2,0),(4,4)]]
-    from sage.rings.finite_rings.integer_mod_ring import Zmod
-    from sage.categories.cartesian_product import cartesian_product
-    return cartesian_product([Zmod(5),Zmod(5)]), D
-
-def CDF_37_4_1():
-    r"""
-    A cyclic `(37,4,1)`-difference family.
-
-    It appears as example 7.28 in [Stinson2004]_ and in the examples VI.16.14 of [DesignHandbook]_.
-
-    EXAMPLES::
-
-        sage: from sage.combinat.designs.database import CDF_37_4_1
-        sage: from sage.combinat.designs.difference_family import is_difference_family
-        sage: G, D = CDF_37_4_1()
-        sage: is_difference_family(G,D,37,4,1)
-        True
-
-    The difference family is available from the constructor::
-
-        sage: _ = designs.difference_family(37,4,1)
-
-    """
-    D = [[0,1,3,24],[0,4,26,32],[0,10,18,30]]
-    from sage.rings.finite_rings.integer_mod_ring import Zmod
-    return Zmod(37), D
-
-def CDF_81_5_1():
-    r"""
-    A cyclic `(81,5,1)`-difference family.
-
-    From the examples VI.16.16 of [DesignHandbook]_.
-
-    EXAMPLES::
-
-        sage: from sage.combinat.designs.database import CDF_81_5_1
-        sage: from sage.combinat.designs.difference_family import is_difference_family
-        sage: G, D = CDF_81_5_1()
-        sage: is_difference_family(G,D,81,5,1)
-        True
-
-    The difference family is available from the constructor::
-
-        sage: _ = designs.difference_family(81,5,1)
-    """
-    D = [[0,1,5,12,26],[0,2,10,40,64],[0,3,18,47,53],[0,9,32,48,68]]
-    from sage.rings.finite_rings.integer_mod_ring import Zmod
-    return Zmod(81), D
-
-def CDF_91_6_1():
-    r"""
-    A cyclic `(91,6,1)`-difference family.
-
-    From the examples VI.16.18 of [DesignHandbook]_.
-
-    EXAMPLES::
-
-        sage: from sage.combinat.designs.database import CDF_91_6_1
-        sage: from sage.combinat.designs.difference_family import is_difference_family
-        sage: G, D = CDF_91_6_1()
-        sage: is_difference_family(G,D,91,6,1)
-        True
-
-    The difference family is available from the constructor::
 
-        sage: _ = designs.difference_family(91,6,1)
-    """
-    D = [[0,1,3,7,25,38], [0,16,21,36,48,62], [0,30,40,63,74,82]]
-    from sage.rings.finite_rings.integer_mod_ring import Zmod
-    return Zmod(91), D
-
-def CDF_121_5_1():
-    r"""
-    A cyclic `(121,5,1)`-difference family.
-
-    From the examples VI.16.16 of [DesignHandbook]_.
-
-    EXAMPLES::
-
-        sage: from sage.combinat.designs.database import CDF_121_5_1
-        sage: from sage.combinat.designs.difference_family import is_difference_family
-        sage: G, D = CDF_121_5_1()
-        sage: is_difference_family(G,D,121,5,1)
-        True
-
-    The difference family is available from the constructor::
-
-        sage: _ = designs.difference_family(121,5,1)
-    """
-    D = [[0,14,26,51,60],[0,15,31,55,59],[0,10,23,52,58],[0,3,36,56,57],
-         [0,7,18,45,50],[0,8,30,47,49]]
-    from sage.rings.finite_rings.integer_mod_ring import Zmod
-    return Zmod(121), D
-
-def CDF_141_5_1():
-    r"""
-    A cyclic `(141,5,1)`-difference family.
-
-    From the examples VI.16.16 of [DesignHandbook]_.
-
-    EXAMPLES::
-
-        sage: from sage.combinat.designs.database import CDF_141_5_1
-        sage: from sage.combinat.designs.difference_family import is_difference_family
-        sage: G, D = CDF_141_5_1()
-        sage: is_difference_family(G,D,141,5,1)
-        True
-
-    The difference family is available from the constructor::
-
-        sage: _ = designs.difference_family(141,5,1)
-    """
-    D = [[0,33,60,92,97], [0,3,45,88,110], [0,18,39,68,139],
-         [0,12,67,75,113], [0,1,15,84,94], [0,7,11,24,30],
-         [0,36,90,116,125]]
-    from sage.rings.finite_rings.integer_mod_ring import Zmod
-    return Zmod(141), D
+DF = {
+##############
+# lambda = 1 #
+##############
+((15,),3,1):[[0,1,4],[0,2,9],['S',0,5,10]],
+((21,),3,1):[[0,1,3],[0,4,12],[0,5,11],['S',0,7,14]],
+((21,),5,1):[[0,1,4,14,16]],
+((25,),3,1):[[0,1,3],[0,4,11],[0,5,13],[0,6,15]],
+((5,5),4,1):[[(0,0),(0,1),(1,0),(2,2)],[(0,0),(0,2),(2,0),(4,4)]],
+((27,),3,1):[[0,1,3],[0,4,11],[0,5,15],[0,6,14],['S',0,9,18]],
+((33,),3,1):[[0,1,3],[0,4,10],[0,5,18],[0,7,19],[0,8,17],['S',0,11,22]],
+((37,),4,1):[[0,1,3,24],[0,4,26,32],[0,10,18,30]],
+((39,),3,1):[[0,1,3],[0,4,18],[0,5,27],[0,6,16],[0,7,15],[0,9,20],['S',0,13,26]],
+((45,),3,1):[[0,1,3],[0,4,10],[0,5,28],[0,7,34],[0,8,32],[0,9,29],[0,12,26],['S',0,15,30]],
+((49,),3,1):[[0,1,3],[0,4,9],[0,6,17],[0,7,23],[0,8,30],[0,10,31],[0,12,36],[0,14,34]],
+((51,),3,1):[[0,1,3],[0,4,9],[0,6,25],[0,7,35],[0,8,22],[0,10,21],[0,12,27],[0,13,31],['S',0,17,34]],
+((55,),3,1):[[0,1,3],[0,4,9],[0,6,16],[0,7,32],[0,8,29],[0,11,42],[0,12,27],[0,14,36],[0,17,37]],
+((57,),3,1):[[0,1,3],[0,4,9],[0,6,13],[0,8,26],[0,10,33],[0,11,32],[0,12,40],[0,14,41],[0,15,35],['S',0,19,38]],
+((63,),3,1):[[0,1,3],[0,4,9],[0,6,13],[0,8,25],[0,10,41],[0,11,44],[0,12,36],[0,14,37],[0,15,43],[0,16,34],['S',0,21,42]],
+((69,),3,1):[[0,1,3],[0,4,9],[0,6,13],[0,8,24],[0,10,38],[0,11,47],[0,12,32],[0,14,40],[0,15,50],[0,17,42],[0,18,39],['S',0,23,46]],
+((75,),3,1):[[0,1,67],[0,2,47],[0,3,41],[0,4,69],[0,5,68],[0,11,55],[0,13,61],[0,15,33],[0,16,52],[0,17,43],[0,19,40],[0,22,51],['S',0,25,50]],
+((81,),3,1):
+    [[0,1,39],[0,2,58],[0,3,34],[0,4,21],[0,5,67],[0,6,15],[0,7,36],
+     [0,8,59],[0,10,63],[0,11,37],[0,12,61],[0,13,48],[0,16,40],['S',0,27,54]],
+((40,),4,1):[[0,1,4,13],[0,2,7,24],[0,6,14,25],['S',0,10,20,30]],
+((49,),4,1):[[0,1,3,8,],[0,4,18,29],[0,6,21,33],[0,9,19,32]],
+((52,),4,1):[[0,1,3,7,],[0,5,19,35],[0,8,20,31],[0,9,24,34],['S',0,13,26,39]],
+((64,),4,1):[[0,1,3,7,],[0,5,18,47],[0,8,33,44],[0,9,19,43],[0,12,26,49],['S',0,16,32,48]],
+((76,),4,1):[[0,1,7,22],[0,2,11,45],[0,3,59,71],[0,4,32,50],[0,10,37,51],[0,13,36,60],['S',0,19,38,57]],
+((81,),5,1):[[0,1,5,12,26],[0,2,10,40,64],[0,3,18,47,53],[0,9,32,48,68]],
+((85,),4,1):[[0,2,41,42],[0,17,32,38],[0,18,27,37],[0,13,29,36],[0,11,31,35],[0,12,26,34,],[0,5,30,33]],
+((65,),5,1):[[0,1,3,31,45],[0,4,10,19,57],[0,5,16,41,48],['S',0,13,26,39,52]],
+((91,),6,1):[[0,1,3,7,25,38], [0,16,21,36,48,62], [0,30,40,63,74,82]],
+((121,),5,1):[[0,14,26,51,60],[0,15,31,55,59],[0,10,23,52,58],[0,3,36,56,57],[0,7,18,45,50],[0,8,30,47,49]],
+((141,),5,1):[[0,33,60,92,97],[0,3,45,88,110],[0,18,39,68,139],[0,12,67,75,113],[0,1,15,84,94],[0,7,11,24,30],[0,36,90,116,125]],
+((161,),5,1):[[0,19,34,73,80],[0,16,44,71,79],[0,12,33,74,78],[0,13,30,72,77],[0,11,36,67,76],[0,18,32,69,75],[0,10,48,68,70],[0,3,29,52,53]],
+((201,),5,1):
+  [[0,1,45,98,100],[0,3,32,65,89],[0,4,54,70,75],[0,6,49,69,91],[0,7,58,81,95],
+   [0,8,34,72,90],[0,9,36,77,96],[0,10,35,83,94],[0,12,40,79,92],[0,15,46,76,93]],
+((217,),7,1):
+    [[0,1,37,67,88,92,149],[0,15,18,65,78,121,137],[0,8,53,79,85,102,107],
+     [0,11,86,100,120,144,190],[0,29,64,165,198,205,207],['S',0,31,62,93,124,155,186]],
+((221,),5,1):
+  [[0,1,24,61,116],[0,3,46,65,113],[0,4,73,89,130],[0,5,77,122,124],[0,6,39,50,118],
+   [0,7,66,81,94],[0,8,38,64,139],[0,9,29,80,107],[0,10,35,93,135],[0,12,34,52,88],[0,14,31,63,84]],
+((3,3,5),5,1):
+  [[(0,1,0),(0,2,0),(1,0,2),(2,0,2),(0,0,1)],
+   [(2,1,0),(1,2,0),(2,2,2),(1,1,2),(0,0,1)],
+   ['S', (0,0,0),(0,0,1),(0,0,2),(0,0,3),(0,0,4)]],
+((11,11),6,1):
+  [[(0,0),(0,3),(0,4),(1,1),(1,7),(4,6)],
+   [(0,0),(0,2),(2,5),(4,7),(6,4),(8,0)],
+   [(0,0),(1,5),(2,0),(4,1),(6,0),(7,2)],
+   [(0,0),(1,0),(3,9),(4,8),(6,1),(9,5)]],
+# the following one is lemma 2.2 in Abel "Some new BIBDs with block size 7"
+((7,37),7,1):
+    [['S',(0,0),(1,0),(2,0),(3,0),(4,0),(5,0),(6,0)],
+     [(0,0),(0,1),(0,6),(1,4),(2,19),(3,25),(6,26)],
+     [(0,0),(0,10),(0,23),(2,3),(4,5),(5,1),(6,28)],
+     [(0,0),(0,8),(0,26),(1,13),(3,10),(4,30),(5,21)],
+     [(0,0),(0,4),(1,25),(1,34),(2,33),(2,35),(4,10)],
+     [(0,0),(0,3),(1,26),(2,7),(2,28),(4,17),(4,34)],
+     [(0,0),(0,30),(1,7),(1,22),(2,1),(4,21),(4,33)]],
+((7,5,5),7,1):
+  [['S',(0,0,0),(1,0,0),(2,0,0),(3,0,0),(4,0,0),(5,0,0),(6,0,0)],
+   [(0,0,0),(1,1,3),(1,4,2),(2,2,2),(2,3,3),(4,2,0),(4,3,0)],
+   [(0,0,0),(1,3,4),(1,2,1),(2,2,3),(2,3,2),(4,0,2),(4,0,3)],
+   [(0,0,0),(1,1,2),(1,4,3),(2,1,1),(2,4,4),(4,0,1),(4,0,4)],
+   [(0,0,0),(1,3,1),(1,2,4),(2,4,1),(2,1,4),(4,1,0),(4,4,0)]],
+
+##############
+# lambda = 2 #
+##############
+((16,),3,2):[[0,1,2],[0,2,8],[0,3,7],[0,4,7],[0,5,10]],
+((28,),3,2):[[0,1,12],[0,2,11],[0,2,12],[0,3,7],[0,3,13],[0,4,9],[0,5,13],[0,6,7],[0,6,14]],
+((40,),3,2):[[0,1,4],[0,1,16],[0,2,7],[0,2,9],[0,3,17],[0,4,17],[0,5,19],[0,6,16],[0,6,18],[0,8,18],[0,8,19],[0,9,20],[0,12,25]],
+((19,),4,2):[[0,1,3,12],[0,1,5,13],[0,4,6,9]],
+((21,),4,3):[[0,2,3,7],[0,3,5,9],[0,1,7,11],[0,2,8,11],[0,1,9,14]],
+((22,),4,2):[[0,4,16,17],[0,12,14,21],[0,14,16,19],['S',0,4,11,15]],
+((31,),4,2):[[0,1,8,11],[0,1,13,17],[0,2,11,14],[0,5,7,13],[0,5,9,15]],
+((34,),4,2):[[0,1,22,24],[0,1,19,25],[0,2,6,29],[0,4,7,20],[0,5,8,20],['S',0,8,17,25]],
+((43,),4,2):[[0,1,6,36],[0,3,18,22],[0,9,11,23],[0,10,12,26],[0,26,27,33],[0,13,35,38],[0,19,28,39,]],
+((46,),4,2):[[0,2,7,10],[0,4,19,32],[0,10,34,35],[0,5,8,24],[0,26,30,39],[0,17,26,32],[0,28,34,45],['S',0,2,23,25]],
+((31,),5,2):[[0,1,3,7,15],[0,3,9,14,21],[0,4,5,13,15,]],
+((35,),5,2):[[0,2,8,12,13],[0,3,18,22,27],[0,17,23,32,33],['S',0,7,14,21,28],['S',0,7,14,21,28]],
+((51,),5,2):[[0,1,14,31,35],[0,1,9,23,33],[0,11,16,18,42],[0,7,13,36,39],[0,4,10,12,15]],
+((71,),5,2):[[1,5,25,54,57],[3,4,15,20,29],[9,12,16,45,60],[27,36,38,48,64],[2,10,37,43,50],[6,8,30,40,58],[18,19,24,32,49]],
+((46,),6,2):[[0,1,3,11,31,35],[0,1,4,10,23,29],[0,2,7,15,32,41]],
+((61,),6,2):[[12,15,28,34,35,59],[1,13,18,47,51,53],[8,10,11,21,29,43],[16,20,25,32,40,50]],
+((43,),7,2):[[0,1,11,19,31,38,40],[0,2,10,16,25,38,42]],
+((3,3,17),9,2):
+   [['S', (0,0,0),(0,1,0),(0,2,0),(1,0,0),(1,1,0),(1,2,0),(2,0,0),(2,1,0),(2,2,0)],
+    ['S', (0,0,0),(0,1,0),(0,2,0),(1,0,0),(1,1,0),(1,2,0),(2,0,0),(2,1,0),(2,2,0)],
+    [(0,0,0),(0,1,1),(0,1,16),(0,2,4),(0,2,13),(1,0,3),(1,0,14),(2,0,5),(2,0,12)],
+    [(0,0,0),(0,1,2),(0,1,15),(0,2,8),(0,2,9),(1,0,6),(1,0,11),(2,0,10),(2,0,7)],
+    [(0,0,0),(0,1,3),(0,1,14),(0,2,12),(0,2,5),(1,0,9),(1,0,8),(2,0,15),(2,0,2)],
+    [(0,0,0),(0,1,6),(0,1,11),(0,2,7),(0,2,10),(1,0,1),(1,0,16),(2,0,13),(2,0,4)]],
+((85,),7,2):[[0,1,11,20,32,35,39],[0,2,6,16,29,50,65],[0,3,9,27,55,72,80],[0,5,7,30,47,48,59]],
+((85,),8,2):[[24,31,39,50,67,68,70,82],[20,49,51,55,56,60,72,81],[9,19,29,37,43,56,59,81]],
+((181,),10, 2):
+    [[1,7,40,42,51,59,113,125,135,151],
+     [19,22,31,35,36,64,74,133,154,156],
+     [10,15,34,47,58,65,83,87,161,164],
+     [12,18,32,52,77,78,142,157,165,172]],
+((64,),7,2):[[0,1,2,4,7,28,52],[0,4,9,21,31,39,53],[0,6,15,23,34,41,54]],
+((5,15),5,2):
+  [[(0,0),(1,10),(1,8),(4,1),(4,2)],
+   [(0,0),(2,5),(2,10),(3,7),(3,13)],
+   [(0,0),(1,10),(1,2),(4,4),(4,8)],
+   [(0,0),(2,5),(2,10),(3,14),(3,11)],
+   [(0,0),(1,4),(1,5),(4,1),(4,8)],
+   [(0,0),(1,1),(1,5),(4,4),(4,2)],
+   [(0,0),(2,7),(2,13),(3,1),(3,4)],
+   ['S',(0,0),(1,0),(2,0),(3,0),(4,0)],
+   ['S',(0,0),(1,0),(2,0),(3,0),(4,0)]],
+
+##############
+# lambda = 3 #
+##############
+
+((21,),6,3):[[0,2,10,15,19,20],[0,3,7,9,10,16]],
+((41,),6,3):[[0,1,10,16,18,37],[0,6,14,17,19,26],[0,2,20,32,33,36],[0,11,12,28,34,38]],
+((51,),6,3):[[15,17,18,27,34,48],[3,17,30,34,42,45],[9,17,24,33,34,39],[3,25,41,43,44,48],[3,5,25,29,43,48]],
+((61,),6,3):[[0,1,9,20,58,34],[0,2,7,18,40,55],[0,4,14,19,36,49],[0,8,11,28,37,38],[0,13,15,16,22,56],[0,26,30,32,44,51]],
+((29,),7,3):[[1,7,16,20,23,24,25],[2,3,11,14,17,19,21]],
+((43,),7,3):[[1,4,11,16,21,35,41],[3,5,12,19,20,33,37],[9,13,14,15,17,25,36]],
+((57,),7,3):[[0,1,11,12,15,35,53],[0,7,17,20,27,29,48],[0,5,18,26,32,49,51],[0,2,6,9,14,41,42]],
+((61,),10,3):[[1,4,18,20,32,35,36,41,42,54],[11,13,14,21,23,28,34,39,43,47]],
+((71,),7,3):[[1,20,30,32,37,45,48],[2,3,19,25,40,60,64],[4,6,9,38,49,50,57],[5,8,12,18,27,29,43],[10,15,16,24,36,54,58]],
+((85,),7,3):
+    [[0,7,23,27,28,31,71],
+     [0,12,22,41,61,74,79],
+     [0,6,11,13,38,42,77],
+     [0,1,7,16,19,27,49],
+     [0,9,26,39,54,56,71],
+     [0,2,3,12,37,53,63]],
+((97,),9,3):[[1,2,25,35,46,58,61,70,90],[3,4,8,38,43,50,69,86,87],[6,12,16,32,53,55,57,75,82],[9,18,24,26,31,34,37,48,64]],
+((49,),9,3):[[0,1,3,5,9,14,19,25,37],[0,2,12,13,16,19,34,41,42]],
+((11,11),10,3):
+  [[(0,1),(0,3),(0,4),(0,5),(0,9),(1,8),(3,2),(4,10),(5,7),(9,6)],
+   [(1,2),(3,6),(4,8),(5,10),(9,7),(10,2),(8,6),(7,8),(6,10),(2,7)],
+   [(1,7),(3,10),(4,6),(5,2),(9,8),(1,4),(3,1),(4,5),(5,9),(9,3)],
+   [(10,10),(8,8),(7,7),(6,6),(2,2),(1,0),(3,0),(4,0),(5,0),(9,0)]],
+
+###############
+# lambda = 4  #
+###############
+
+((22,),7,4):[[0,2,6,8,9,10,13],[0,3,5,6,12,13,17]],
+((29,),8,4):[[0,1,7,16,20,23,24,25],[0,2,3,11,14,17,19,21]],
+((71,),8,4):[[0,1,20,30,32,37,45,48],[0,2,3,19,25,40,60,64],[0,4,6,9,38,49,50,57],[0,5,8,12,18,27,29,43],[0,10,15,16,24,36,54,58]],
+((43,),8,4):[[0,1,4,11,16,21,35,41],[0,3,5,12,19,20,33,37],[0,9,13,14,15,17,25,36]],
+((46,),10,4):[[3,7,13,16,23,24,25,28,30,42],[2,10,12,18,25,34,40,43,44,45]],
+((55,),9,4):[[0,4,21,25,26,42,45,53,54],[0,6,8,25,37,39,45,48,52],[2,5,6,13,15,20,25,39,45]],
+((67,),12,4):[[1,8,23,25,28,29,31,37,47,54,55,64],[3,20,25,32,36,39,44,45,54,55,57,59]],
+
+##############
+# lambda = 5 #
+##############
+
+((13,),5,5):[[0,1,2,4,8],[0,1,3,6,12],[0,2,5,6,10]],
+((17,),5,5):[[0,1,4,13,16],[0,3,5,12,14],[0,2,8,9,15],[0,6,7,10,11]],
+((21,),6,5):[[0,2,6,12,15,16],[0,3,6,7,11,19],[0,7,15,16,17,18],['S',0,2,7,9,14,16]],
+((22,),6,5):[[0,1,2,5,10,13],[0,1,5,6,8,15],[0,2,3,6,16,18],['S',0,2,6,11,13,17]],
+((28,),6,5):[[0,4,7,8,16,21],[5,7,8,9,14,20],[7,12,14,16,17,25],[1,4,7,13,14,24],['S',2,4,8,16,18,22]],
+((33,),5,5):[[0,2,3,7,25],[0,3,13,14,29],[0,4,5,12,13],[0,2,12,16,26],[0,3,12,20,31],[3,9,12,15,27],[0,8,13,14,31],[0,2,7,13,29]],
+((33,),6,5):[[0,3,12,17,18,28],[0,2,3,16,28,29],[0,16,20,26,28,30],[0,2,3,12,16,27],[0,6,20,21,28,30],['S',0,4,11,15,22,26]],
+((37,),10, 5):[[0,1,7,9,10,12,16,26,33,34],[0,2,14,15,18,20,24,29,31,32]],
+((39,),6,5):[[0,3,4,17,19,32],[0,1,5,12,30,36],[0,3,8,9,25,27],[0,7,10,12,17,21],[0,16,18,19,27,35],[0,2,18,27,28,33],['S',0,6,13,19,26,32]],
+((45,),11, 5):[[1,3,7,10,22,25,30,35,37,38,44],[0,2,3,14,22,26,27,28,31,32,38]],
+((46,),10, 5):[[0,4,6,11,12,15,24,25,28,42],[0,2,5,7,8,9,14,24,34,35],['S',0,2,12,32,40,23,25,35,9,17]],
+((55,),10, 5):[[0,5,11,15,20,22,25,33,44,45],[3,7,8,10,31,37,39,45,46,49],[3,7,8,10,31,37,39,45,46,49]],
+((67,),11, 5):[[1,9,14,15,22,24,25,40,59,62,64],[2,13,18,28,30,44,48,50,51,57,61],[4,21,26,29,33,35,36,47,55,56,60]],
+((73,),10,5):[[0,1,2,4,8,16,32,37,55,64],[0,5,7,10,14,20,28,39,40,56],[0,25,27,35,49,50,54,61,67,70],[0,11,15,21,22,30,42,44,47,60]],
+
+###############
+# lambda >= 6 #
+###############
+
+((11,),4,6):[[0,1,8,9],[0,2,5,7],[0,1,4,5],[0,2,3,5],[0,4,5,9]],
+((15,),4,6):[[0,1,2,3],[0,2,4,6],[0,4,8,12],[0,8,1,9],[3,6,9,12],[0,1,5,10],[0,2,5,10]],
+((15,),5,6):[[0,1,2,3,6],[0,2,4,7,8],[0,2,4,9,10],[0,3,6,10,11],['S',0,3,6,9,12]],
+((29,),8,6):[[0,5,10,11,12,13,16,20],[0,8,10,12,17,22,23,26],[0,4,5,11,13,23,25,26]],
+((46,),10,6):[[0,2,11,13,21,22,30,33,34,40],[0,2,6,7,22,23,28,32,35,38],[0,2,4,7,8,9,12,23,26,41]],
+((53,),13,6):[[1,10,13,15,16,24,28,36,42,44,46,47,49],[2,3,19,20,26,30,31,32,35,39,41,45,48]],
+((67,),12,6):[[0,1,9,14,15,22,24,25,40,59,62,64],[0,2,13,18,28,30,44,48,50,51,57,61],[0,4,21,26,29,33,35,36,47,55,56,60]],
+
+((53,),14,7):[[0,1,10,13,15,16,24,28,36,42,44,46,47,49],[0,2,3,19,20,26,30,31,32,35,39,41,45,48]],
+((61,),15,7):[[0,1,3,4,8,10,13,22,30,35,44,45,46,50,58],[0,1,3,5,13,18,29,34,35,37,41,43,44,51,55]],
+((5,5),7,7):
+  [[(0,0),(0,1),(0,4),(1,1),(1,2),(4,3),(4,4)],
+   [(0,0),(1,0),(1,3),(2,3),(3,2),(4,0),(4,2)],
+   [(0,0),(0,2),(0,3),(2,2),(2,4),(3,1),(3,3)],
+   [(0,0),(1,4),(2,0),(2,1),(3,0),(3,4),(4,1)]],
+
+((22,),8,8):[[0,1,5,7,13,17,20,21],[0,2,7,11,13,14,16,17],[0,3,4,12,14,15,17,21]],
+((34,),12,8):[[0,5,9,14,15,17,20,25,26,27,28,30],[0,6,7,10,13,17,18,20,22,24,25,26]],
+
+((21,),10,9):[[0,1,2,3,4,7,8,11,14,16],[0,6,7,9,11,12,15,16,17,19]],
+
+((34,),12,10):[[0,2,3,4,8,9,11,13,14,24,27,30],[0,2,6,7,8,11,13,14,22,25,26,32],['S',0,2,10,18,22,32,17,19,27,1,5,15]],
+((43,),15,10):[[1,3,6,13,18,21,22,25,26,27,33,35,36,38,40],[9,10,11,13,16,17,19,23,26,27,28,33,35,38,39]],
+((7,7),21,10):
+   [['S',(0,1),(0,2),(0,4),(1,1),(1,2),(1,4),(2,1),(2,2),(2,4),(3,1),(3,2),
+         (3,4),(4,1),(4,2),(4,4),(5,1),(5,2),(5,4),(6,1),(6,2),(6,4)],
+    [(1,0),(1,1),(1,2),(1,4),(2,0),(2,1),(2,2),(2,4),(4,0),(4,1),(4,2),
+    (4,4),(3,3),(3,5),(3,6),(5,3),(5,5),(5,6),(6,3),(6,5),(6,6)]],
+
+((22,),8,12):
+    [[1,2,3,5,6,9,15,18],
+     [1,2,3,5,8,9,10,15],
+     [1,3,4,9,13,18,19,21],
+     ['S',2,4,6,12,13,15,17,1],
+     ['S',2,4,8,12,13,15,19,1],
+     ['S',2,4,8,16,13,15,19,5]],
+
+((21,),8,14):[[0,9,10,13,14,15,18,19],[0,1,4,7,9,15,16,18],[0,1,2,4,6,14,15,16],[0,1,3,4,8,14,16,18],[0,1,4,9,11,12,14,16]],
+}
 
-def CDF_161_5_1():
+def DF_from_database(v, k, l, short_blocks=True, check=False):
     r"""
-    A cyclic `(161,5,1)`-difference family.
-
-    From the examples VI.16.16 of [DesignHandbook]_.
+    Return a ``(v,k,l)``-difference family from the database or ``False`` if
+    there is none.
 
-    EXAMPLES::
+    INPUT:
 
-        sage: from sage.combinat.designs.database import CDF_161_5_1
-        sage: from sage.combinat.designs.difference_family import is_difference_family
-        sage: G, D = CDF_161_5_1()
-        sage: is_difference_family(G,D,161,5,1)
-        True
+    - ``v``, ``k``, ``l`` -- the parameters of the difference family
 
-    The difference family is available from the constructor::
+    - ``short_blocks`` -- whether short blocks are allowed (default to ``True``)
 
-        sage: _ = designs.difference_family(161,5,1)
-    """
-    D = [[0,19,34,73,80], [0,16,44,71,79], [0,12,33,74,78],
-         [0,13,30,72,77], [0,11,36,67,76], [0,18,32,69,75],
-         [0,10,48,68,70], [0,3,29,52,53]]
-    from sage.rings.finite_rings.integer_mod_ring import Zmod
-    return Zmod(161), D
-
-def CDF_201_5_1():
-    r"""
-    A cyclic `(201,5,1)`-difference family.
-
-    From the examples VI.16.16 of [DesignHandbook]_.
+    - ``check`` -- mainly for debug purpose (default to ``False``)
 
     EXAMPLES::
 
-        sage: from sage.combinat.designs.database import CDF_201_5_1
-        sage: from sage.combinat.designs.difference_family import is_difference_family
-        sage: G, D = CDF_201_5_1()
-        sage: is_difference_family(G,D,201,5,1)
-        True
-
-    The difference family is available from the constructor::
+        sage: from sage.combinat.designs.database import DF_from_database
+        sage: DF_from_database(15,3,1)
+        (Ring of integers modulo 15, [[0, 1, 4], [0, 2, 9], [0, 5, 10]])
+        sage: DF_from_database(15,3,1,short_blocks=False)
+        False
 
-        sage: _ = designs.difference_family(201,5,1)
+        sage: from sage.combinat.designs.database import DF, DF_from_database
+        sage: for v,k,l in DF:
+        ....:     df = DF_from_database(v,k,l,check=True)
     """
-    D = [[0,1,45,98,100], [0,3,32,65,89], [0,4,54,70,75],
-         [0,6,49,69,91], [0,7,58,81,95], [0,8,34,72,90],
-         [0,9,36,77,96], [0,10,35,83,94], [0,12,40,79,92],
-         [0,15,46,76,93]]
+    from sage.misc.misc_c import prod
+    from sage.rings.integer import Integer
+    from difference_family import block_stabilizer, is_difference_family
     from sage.rings.finite_rings.integer_mod_ring import Zmod
-    return Zmod(201), D
-
-def CDF_221_5_1():
-    r"""
-    A cyclic `(221,5,1)`-difference family.
-
-    From the examples VI.16.16 of [DesignHandbook]_.
+    from sage.categories.cartesian_product import cartesian_product
 
-    EXAMPLES::
+    if isinstance(v,(int,Integer)):
+        for (vv,kk,ll) in DF:
+            if kk == k and ll == l and prod(vv) == v:
+                if not short_blocks and any(b[0] == 'S' for b in DF[vv,k,l]):
+                    continue
+                v = vv
+                break
+        else:
+            return False
+    elif (v,k,l) not in DF or (not short_blocks and any(b[0] == 'S' for b in DF[v,k,l])):
+        return False
+
+    if len(v) == 1:
+        A = Zmod(v[0])
+    else:
+        A = cartesian_product([Zmod(i) for i in v])
+
+    blocks = DF[v,k,l]
+
+    df = []
+    short_blocks = False
+    for b in blocks:
+        if b[0] == 'S':
+            short_blocks = True
+            b = [A(i) for i in b[1:]]
+            if check and len(block_stabilizer(A,b)) == 1:
+                print "short block expected but get full with ({},{},{})\n {}".format(v,k,l,b)
+        else:
+            b = [A(i) for i in b]
+            if check and len(block_stabilizer(A,b)) > 1:
+                print "full block expected but get short with ({},{},{})\n  {}".format(v,k,l,b)
 
-        sage: from sage.combinat.designs.database import CDF_221_5_1
-        sage: from sage.combinat.designs.difference_family import is_difference_family
-        sage: G, D = CDF_221_5_1()
-        sage: is_difference_family(G,D,221,5,1)
-        True
+        df.append(b)
 
-    The difference family is available from the constructor::
+    if check and not is_difference_family(A, df, v=prod(v), k=k, l=l, short_blocks=short_blocks):
+        is_difference_family(A,df,v=prod(v),k=k,l=l,short_blocks=short_blocks,verbose=True)
+        raise ValueError("ERROR: pb with ({},{},{})".format(v,k,l))
 
-        sage: _ = designs.difference_family(221,5,1)
-    """
-    D = [[0,1,24,61,116], [0,3,46,65,113], [0,4,73,89,130],
-         [0,5,77,122,124], [0,6,39,50,118], [0,7,66,81,94],
-         [0,8,38,64,139], [0,9,29,80,107], [0,10,35,93,135],
-         [0,12,34,52,88], [0,14,31,63,84]]
-    from sage.rings.finite_rings.integer_mod_ring import Zmod
-    return Zmod(221), D
+    return A,df
 
-# Index of the (right now cyclic or Abelian) difference families constructions
-#
-# Associates to triple (v,k,lambda) a function that return a
-# (n,k,lambda)-difference family.
-#
-# This dictionary is used by designs.difference_family
-
-DF_constructions = {
-    (21,5,1): CDF_21_5_1,
-    (25,4,1): ADF_5x5_4_1,
-    (37,4,1): CDF_37_4_1,
-    (81,5,1): CDF_81_5_1,
-    (91,6,1): CDF_91_6_1,
-    (121,5,1): CDF_121_5_1,
-    (141,5,1): CDF_141_5_1,
-    (161,5,1): CDF_161_5_1,
-    (201,5,1): CDF_201_5_1,
-    (221,5,1): CDF_221_5_1,
-}
 
 def RBIBD_120_8_1():
     r"""
diff --git a/src/sage/combinat/designs/difference_family.py b/src/sage/combinat/designs/difference_family.py
index 17736d94a05..386ebcbbfe7 100644
--- a/src/sage/combinat/designs/difference_family.py
+++ b/src/sage/combinat/designs/difference_family.py
@@ -46,18 +46,115 @@ def group_law(G):
     else:
         raise ValueError("%s does not seem to be a group"%G)
 
-def is_difference_family(G, D, v=None, k=None, l=None, verbose=False):
+def block_stabilizer(G, B):
     r"""
-    Check wether ``D`` forms a difference family in ``G``.
+    Compute the stabilizer of the block ``B`` in the group additive group ``G``.
+
+    EXAMPLES::
+
+        sage: from sage.combinat.designs.difference_family import block_stabilizer
+
+        sage: Z8 = Zmod(8)
+        sage: block_stabilizer(Z8, [Z8(0),Z8(2),Z8(4),Z8(6)])
+        [0, 2, 4, 6]
+        sage: block_stabilizer(Z8, [Z8(0),Z8(2)])
+        [0]
+
+        sage: C = cartesian_product([Zmod(4),Zmod(3)])
+        sage: block_stabilizer(C, [C((0,0)),C((2,0)),C((0,1)),C((2,1))])
+        [(0, 0), (2, 0)]
+
+        sage: b = map(Zmod(45),[1, 3, 7, 10, 22, 25, 30, 35, 37, 38, 44])
+        sage: block_stabilizer(Zmod(45),b)
+        [0]
+    """
+    b0 = -B[0]
+    S = []
+    for b in B:
+        # fun: if we replace +(-b) with -b it completely fails!!
+        if all(b + b0 + c in B for c in B):
+            S.append(b+b0)
+    return S
+
+def orbit_representatives(G,B):
+    r"""
+    Return the orbits of the additive group ``G`` that intersects the set ``B``.
+
+    EXAMPLES::
+
+        sage: from sage.combinat.designs.difference_family import (
+        ....:    block_stabilizer, orbit_representatives)
+
+        sage: Z8 = Zmod(8)
+        sage: b = map(Z8, [0,2,4,6])
+        sage: S = block_stabilizer(Z8,b)
+        sage: orbit_representatives(S, b)
+        [0]
+
+        sage: b = map(Z8, [0,1,4,5])
+        sage: S = block_stabilizer(Z8,b)
+        sage: orbit_representatives(S, b)
+        [0, 1]
+
+        sage: C = cartesian_product([Zmod(4),Zmod(3)])
+        sage: b = [C((0,0)),C((2,0)),C((0,1)),C((2,1))]
+        sage: S = block_stabilizer(C,b)
+        sage: orbit_representatives(S, b)
+        [(0, 1), (2, 0)]
+    """
+    if len(G) == 1:
+        return B
+
+    o = []
+    B = set(B)
+    while B:
+        b = B.pop()
+        o.append(b)
+        for g in G:
+            if not g.is_zero():
+                B.remove(g+b)
+    return o
+
+def partial_differences(G, B):
+    r"""
+    Compute the partial differences of the block ``B``.
+
+    EXAMPLES::
+
+        sage: from sage.combinat.designs.difference_family import partial_differences
+
+        sage: b = map(Zmod(8),[0,2,4,6])
+        sage: partial_differences(Zmod(8),b) 
+        [2, 4, 6]
+
+        sage: b = map(Zmod(8),[0,1,4,5])
+        sage: partial_differences(Zmod(8),b)
+        [1, 3, 4, 4, 5, 7]
+
+        sage: b = [1, 3, 7, 10, 22, 25, 30, 35, 37, 38, 44]
+        sage: b = map(Zmod(45),b)
+        sage: d = partial_differences(Zmod(45),b)
+        sage: len(d)
+        110
+    """
+    stab = block_stabilizer(G,B)
+    o = orbit_representatives(stab,B)
+    return sorted(b + (-c) + g for b in o for c in o for g in stab if b+g != c)
+
+def is_difference_family(G, D, v=None, k=None, l=None, short_blocks=True, verbose=False):
+    r"""
+    Check wether ``D`` forms a difference family in the additive Abelian group ``G``.
 
     INPUT:
 
-    - ``G`` - Abelian group of cardinality ``v``
+    - ``G`` - Additive Abelian group of cardinality ``v``
 
     - ``D`` - a set of ``k``-subsets of ``G``
 
     - ``v``, ``k`` and ``l`` - optional parameters of the difference family
 
+    - ``short_blocks`` - whether short blocks are allowed (default is ``True``)
+
     - ``verbose`` - whether to print additional information
 
     .. SEEALSO::
@@ -75,7 +172,16 @@ def is_difference_family(G, D, v=None, k=None, l=None, verbose=False):
         sage: G = Zmod(41)
         sage: D = [[0,1,4,11,29],[0,2,8,17,21]]
         sage: is_difference_family(G, D, verbose=True)
-        the element 28 in G is obtained more than 1 times
+        Too less:
+          5 is obtained 0 times in blocks []
+          14 is obtained 0 times in blocks []
+          27 is obtained 0 times in blocks []
+          36 is obtained 0 times in blocks []
+        Too much:
+          4 is obtained 2 times in blocks [0, 1]
+          13 is obtained 2 times in blocks [0, 1]
+          28 is obtained 2 times in blocks [0, 1]
+          37 is obtained 2 times in blocks [0, 1]
         False
         sage: D = [[0,1,4,11,29],[0,2,8,17,22]]
         sage: is_difference_family(G, D)
@@ -95,19 +201,45 @@ def is_difference_family(G, D, v=None, k=None, l=None, verbose=False):
         sage: is_difference_family(G, D)
         True
 
-    The function also supports multiplicative groups (non necessarily Abelian)::
+    An example with a short block (i.e. considering restricted difference when a
+    block has a non trivial stabilizer)::
+
+        sage: G = Zmod(15)
+        sage: D = [[0,1,4],[0,2,9],[0,5,10]]
+        sage: is_difference_family(G,D,verbose=True)
+        It is a (15,3,1)-difference family
+        True
+        sage: is_difference_family(G,D,short_blocks=False,verbose=True)
+        bk(k-1) is not 0 mod (v-1)
+        False
+
+    The function also supports multiplicative groups (non necessarily Abelian).
+    In that case, the argument ``short_blocks`` must be set to ``False``::
 
         sage: G = DihedralGroup(8)
         sage: x,y = G.gens()
         sage: D1 = [[1,x,x^4], [1,x^2, y*x], [1,x^5,y], [1,x^6,y*x^2], [1,x^7,y*x^5]]
-        sage: is_difference_family(G, D1, 16, 3, 2)
+        sage: is_difference_family(G, D1, 16, 3, 2, short_blocks=False)
         True
+        sage: is_difference_family(G, D1, 16, 3, 2)
+        Traceback (most recent call last):
+        ...
+        ValueError: short blocks are meaningful only for Abelian group that we assume
+        to be additive
     """
+    import operator
+
+    identity, mul, inv = group_law(G)
+    if short_blocks and not (mul is operator.add or inv is operator.neg):
+        raise ValueError("short blocks are meaningful only for Abelian group that we assume to be additive")
+
+    Glist = list(G)
+
     if v is None:
-        v = G.cardinality()
+        v = len(Glist)
     else:
         v = int(v)
-        if G.cardinality() != v:
+        if len(Glist) != v:
             if verbose:
                 print "G must have cardinality v (=%d)"%v
             return False
@@ -119,52 +251,112 @@ def is_difference_family(G, D, v=None, k=None, l=None, verbose=False):
 
     b = len(D)
 
-    if any(len(d) != k for d in D):
-        if verbose:
-            print "each element of D must have cardinality k (=%d)"%k
-        return False
-
-    if l is None:
-        if (b*k*(k-1)) % (v-1) != 0:
-            if verbose:
-                print "bk(k-1) is not 0 mod (v-1)"
-            return False
-        l = b*k*(k-1) // (v-1)
-    else:
-        l = int(l)
-        if b*k*(k-1) != l*(v-1):
+    for d in D:
+        if len(d) != k:
             if verbose:
-                print "the relation bk(k-1) == l(v-1) is not satisfied with b=%d, k=%d, l=%d, v=%d"%(b,k,l,v)
+                print "the block {} does not have length {}".format(d,k)
             return False
 
-    identity, op, inv = group_law(G)
+    if short_blocks:
+        nb_diff = 0
+        stab_sizes = []
+        for d in D:
+            s = len(block_stabilizer(G,map(G,d)))
+            stab_sizes.append(s)
+            nb_diff += k*(k-1) / s
+        if l is None:
+            if nb_diff % (v-1) != 0:
+                if verbose:
+                    print "sum(1/s_i) k*(k-1) = {} is not a multiple of (v-1) = {} (stabilizer sizes {})".format(
+                            nb_diff, v-1, stab_sizes)
+                return False
+            l = nb_diff // (v-1)
+        else:
+            if nb_diff != l*(v-1):
+                if verbose:
+                    print ("the relation sum(1/s_i) *k*(k-1) == l*(v-1) is not "
+                    "satisfied (where the s_i are the cardinality of the stabilizer "
+                    "for each blocks)")
+                return False
+
+    else:
+        if l is None:
+            if (b*k*(k-1)) % (v-1) != 0:
+                if verbose:
+                    print "bk(k-1) is not 0 mod (v-1)"
+                return False
+            l = b*k*(k-1) // (v-1)
+        else:
+            l = int(l)
+            if b*k*(k-1) != l*(v-1):
+                if verbose:
+                    print "the relation bk(k-1) == l(v-1) is not satisfied with b=%d, k=%d, l=%d, v=%d"%(b,k,l,v)
+                return False
 
     # now we check that every non-identity element of G occurs exactly l-time
     # as a difference
-    counter = {g: 0 for g in G}
+    counter = {g: 0 for g in Glist}
+    where   = {g: [] for g in Glist}
     del counter[identity]
 
-    for d in D:
-        dd = map(G,d)
-        for i in xrange(k):
-            for j in xrange(k):
-                if i == j:
-                    continue
-                g = op(dd[i], inv(dd[j]))
-                if g == identity:
+    if short_blocks:
+        for i,d in enumerate(D):
+            dd = map(G,d)
+            for g in partial_differences(G,dd):
+                if g.is_zero():
                     if verbose:
-                        print "two identical elements in the same block"
+                        print "two identical elements in block {}".format(dd)
                     return False
+                where[g].append(i)
                 counter[g] += 1
-                if counter[g] > l:
-                    if verbose:
-                        print "the element %s in G is obtained more than %s times"%(g,l)
-                    return False
+    else:
+        for i,d in enumerate(D):
+            dd = map(G,d)
+            for ix,x in enumerate(dd):
+                for iy,y in enumerate(dd):
+                    if ix == iy:
+                        continue
+                    if x == y:
+                        if verbose:
+                            print "two identical elements in block {}".format(dd)
+                        return False
+                    g = mul(x,inv(y))
+                    where[g].append(i)
+                    counter[g] += 1
+
+    too_less = []
+    too_much = []
+    for g in Glist:
+        if g == identity:
+            continue
+        if counter[g] < l:
+            if verbose:
+                too_less.append(g)
+            else:
+                return False
+        if counter[g] > l:
+            if verbose:
+                too_much.append(g)
+            else:
+                return False
+
+    if too_less:
+        print "Too less:"
+        for g in too_less:
+            print "  {} is obtained {} times in blocks {}".format(
+                        g,counter[g],where[g])
+    if too_much:
+        print "Too much:"
+        for g  in too_much:
+            print "  {} is obtained {} times in blocks {}".format(
+                        g,counter[g],where[g])
+    if too_less or too_much:
+        return False
+
     if verbose:
         print "It is a ({},{},{})-difference family".format(v,k,l)
     return True
 
-
 def singer_difference_set(q,d):
     r"""
     Return a difference set associated to the set of hyperplanes in a projective
@@ -248,9 +440,9 @@ def singer_difference_set(q,d):
 
     return Zmod((q**(d+1)-1)//(q-1)), [powers]
 
-def difference_family(v, k, l=1, existence=False, check=True):
+def difference_family(v, k, l=1, short_blocks=True, existence=False, check=True):
     r"""
-    Return a (``k``, ``l``)-difference family on an Abelian group of size ``v``.
+    Return a (``k``, ``l``)-difference family on an Abelian group of cardinality ``v``.
 
     Let `G` be a finite Abelian group. For a given subset `D` of `G`, we define
     `\Delta D` to be the multi-set of differences `\Delta D = \{x - y; x \in D,
@@ -304,9 +496,9 @@ def difference_family(v, k, l=1, existence=False, check=True):
         sage: for q in islice(prime_power_mod(1,30), 60):
         ....:     l6[designs.difference_family(q,6,existence=True)].append(q)
         sage: l6[True]
-        [31, 151, 181, 211, ...,  3061, 3121, 3181]
+        [31, 121, 151, 181, 211, ...,  3061, 3121, 3181]
         sage: l6[Unknown]
-        [61, 121]
+        [61]
         sage: l6[False]
         []
 
@@ -331,34 +523,56 @@ def difference_family(v, k, l=1, existence=False, check=True):
         ....:                 _ = designs.difference_family(v,k,l)
         ....:     if constructions:
         ....:         print "%2d: %s"%(v, ', '.join('(%d,%d)'%(k,l) for k,l in constructions))
-         4: (3,2)
-         5: (4,3)
-         7: (3,2), (6,5)
-         8: (7,6)
-         9: (4,3), (8,7)
-        11: (5,2), (5,4)
-        13: (3,2), (4,3), (6,5)
-        15: (7,3)
-        16: (3,2), (5,4)
-        17: (4,3), (8,7)
-        19: (3,2), (6,5), (9,4), (9,8)
-        25: (3,2), (4,3), (6,5), (8,7)
-        29: (4,3), (7,6)
-        31: (3,2), (5,4), (6,5)
-        37: (3,2), (4,3), (6,5), (9,2), (9,8)
-        41: (4,3), (5,4), (8,7)
-        43: (3,2), (6,5), (7,6)
-        49: (3,2), (4,3), (6,5), (8,7)
-        53: (4,3)
-        61: (3,2), (4,3), (5,4), (6,5)
-        64: (3,2), (7,6), (9,8)
-        67: (3,2), (6,5)
-        71: (5,4), (7,6)
-        73: (3,2), (4,3), (6,5), (8,7), (9,8)
-        79: (3,2), (6,5)
-        81: (4,3), (5,4), (8,7)
-        89: (4,3), (8,7)
-        97: (3,2), (4,3), (6,5), (8,7)
+         2: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+         3: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+         4: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+         5: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+         7: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+         8: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+         9: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+        11: (3,2), (4,3), (4,6), (5,2), (5,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+        13: (3,2), (4,3), (5,4), (5,5), (6,5), (7,6), (8,7), (9,8)
+        15: (4,6), (5,6), (7,3)
+        16: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+        17: (3,2), (4,3), (5,4), (5,5), (6,5), (7,6), (8,7), (9,8)
+        19: (3,2), (4,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,4), (9,5), (9,6), (9,7), (9,8)
+        21: (4,3), (6,3), (6,5)
+        22: (4,2), (6,5), (7,4), (8,8)
+        23: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+        25: (3,2), (4,3), (5,4), (6,5), (7,6), (7,7), (8,7), (9,8)
+        27: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+        28: (3,2), (6,5)
+        29: (3,2), (4,3), (5,4), (6,5), (7,3), (7,6), (8,4), (8,6), (8,7), (9,8)
+        31: (3,2), (4,2), (4,3), (5,2), (5,4), (6,5), (7,6), (8,7), (9,8)
+        32: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+        33: (5,5), (6,5)
+        34: (4,2)
+        35: (5,2), (8,4)
+        37: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,2), (9,3), (9,8)
+        39: (6,5)
+        40: (3,2)
+        41: (3,2), (4,3), (5,4), (6,3), (6,5), (7,6), (8,7), (9,8)
+        43: (3,2), (4,2), (4,3), (5,4), (6,5), (7,2), (7,3), (7,6), (8,4), (8,7), (9,8)
+        46: (4,2), (6,2)
+        47: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+        49: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,3), (9,8)
+        51: (5,2), (6,3)
+        53: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+        55: (9,4)
+        57: (7,3)
+        59: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+        61: (3,2), (4,3), (5,4), (6,2), (6,3), (6,5), (7,6), (8,7), (9,8)
+        64: (3,2), (4,3), (5,4), (6,5), (7,2), (7,6), (8,7), (9,8)
+        67: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+        71: (3,2), (4,3), (5,2), (5,4), (6,5), (7,3), (7,6), (8,4), (8,7), (9,8)
+        73: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+        75: (5,2)
+        79: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+        81: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+        83: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+        85: (7,2), (7,3), (8,2)
+        89: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,8)
+        97: (3,2), (4,3), (5,4), (6,5), (7,6), (8,7), (9,3), (9,8)
 
     TESTS:
 
@@ -387,28 +601,23 @@ def difference_family(v, k, l=1, existence=False, check=True):
 
     .. TODO::
 
-        There is a slightly more general version of difference families where
-        the stabilizers of the blocks are taken into account. A block is *short*
-        if the stabilizer is not trivial. The more general version is called a
-        *partial difference family*. It is still possible to construct BIBD from
-        this more general version (see the chapter 16 in the Handbook
-        [DesignHandbook]_).
-
         Implement recursive constructions from Buratti "Recursive for difference
         matrices and relative difference families" (1998) and Jungnickel
         "Composition theorems for difference families and regular planes" (1978)
     """
-    if (l*(v-1)) % (k*(k-1)) != 0:
+    if not short_blocks and (l*(v-1)) % (k*(k-1)) != 0:
         if existence:
             return False
         raise EmptySetError("A (v,%d,%d)-difference family may exist only if %d*(v-1) = mod %d"%(k,l,l,k*(k-1)))
 
     from block_design import are_hyperplanes_in_projective_geometry_parameters
-    from database import DF_constructions
-    if (v,k,l) in DF_constructions:
+    from database import DF_from_database
+
+    G_df = DF_from_database(v,k,l,short_blocks=short_blocks)
+    if G_df:
         if existence:
             return True
-        return DF_constructions[(v,k,l)]()
+        return G_df
 
     e = k*(k-1)
     t = l*(v-1) // e  # number of blocks
@@ -506,7 +715,7 @@ def difference_family(v, k, l=1, existence=False, check=True):
             return Unknown
         raise NotImplementedError("No constructions for these parameters")
 
-    if check and not is_difference_family(G,D,verbose=False):
+    if check and not is_difference_family(G,D,short_blocks=short_blocks,verbose=False):
         raise RuntimeError
 
     return G, D