What is the shortest brainfuck quine?

[void4]

(apart from the empty program of course)

The shortest I could find was this 400 byte beauty:

Quine	Length	Steps	Cells	Symbols	Source
True	6953	137819	123	Counter({'+': 6758, '>': 138, '<': 20, '[': 10, '-': 10, ']': 10, '.': 7})	>>>>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++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True	3380	585943	802	Counter({'+': 2297, '>': 863, '-': 66, '<': 65, '[': 41, ']': 41, '.': 7})	>>>>>+++++>+++++>+++++>+++++>+++++>>++++++>++++>+++++++>++++>++++++>++++>+++++++>++++>++++++>++++>+++++++>+++++>++++++>++++>++++++>++>+++++++>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+++++>++++++>++++>++>++>++>++>++>++>++>++>++>++>++>++>++>++>++>++>++>+++++>++++>++++>+>+++++>+++++>++>++++++>++++>+>+>+++++>++++>++++>+>+++++>+++++>++>++++++>++++>+>+++++>++++>++++>+>+++++>+++++>++>++++++>++++>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+++++>++++>++++>+>+++++>+++++>++>++++++>++++>+>+>+++++>++++>++++>+>+++++>+++++>++>++++++>++++>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+++++>++++>++++>+>+++++>+++++>++>++++++>++++>+>+>+++++>++++>++++>+>+++++>+++++>++>++++++>++>+++++++>+++++++>+++++++>+++++++>+++++++>+++++++>+++++++>+++++++>++++>+++>++++++>++>+++++++>+++++>+++++>+++++++>++++>++++>++++>++++++>++++>+++++++>+++++>++++++>++++++>++++>++++>+>+++++>+++++>++++>++++++>++>+++++++>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+++>++++++>++>+++++++>+++++>++>+++++++>++++++>++>+++++++>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+++>++++++>++>+++++++>+++++>+++++++>++++++>++>+++++++>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+++>++++++>++>+++++++>+++++>+++++>+++++>++++++>++++++>++++>++++>+>+++++>+++++>++++>++++++>++>+++++++>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+++>++++++>++>+++++++>+++++>++>+++++++>++++++>++>+++++++>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+++>++++++>++>+++++++>+++++>+++++++>++++>++++>++++>++++++>++++>+++++++>+++++>++++++>++++>++++++>++>+++++++>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+++++>++++++>++++>++>++>++>++>++>++>++>++>++>++>++>++>++>++>++>++>++>+++++>++++>++++>+>+++++>+++++>++>++++++>++++>+>+>+++++>++++>++++>+>+++++>+++++>++>++++++>++++>+>+++++>++++>++++>+>+++++>+++++>++>++++++>++++>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+++++>++++>++++>+>+++++>+++++>++>++++++>++++>+>+>+++++>++++>++++>+>+++++>+++++>++>++++++>++++>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+>+++++>++++>++++>+>+++++>+++++>++>++++++>++++>+>+>+++++>++++>++++>+>+++++>+++++>++>++++++>++>+++++++>+++++++>+++++++>+++++++>+++++++>+++++++>+++++++>+++++++>++++>+++>++++++>++>+++++++>+++++>+++++>+++++++>[<]<[<]<[<]>[<[-]++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>[<-----------------><<+>>-[<++><<+>>-[<+><<+>>-[<++++++++++++++><<+>>-[<++><<+>>-[<+++++++++++++++++++++++++++++><<+>>-[<++><<+>>-[-]]]]]]]]<.[-]>>]<<<[<]>[[<<+>><[-]+++++++++++++++++++++++++++++++++++++++++++.[-]>-][-]++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++.[-]>][-]++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++.[-]>>>[[<<+>><[-]+++++++++++++++++++++++++++++++++++++++++++.[-]>-][-]++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++.[-]>]<<<[<]>[<[-]++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++>[<-----------------><<+>>-[<++><<+>>-[<+><<+>>-[<++++++++++++++><<+>>-[<++><<+>>-[<+++++++++++++++++++++++++++++><<+>>-[<++><<+>>-[-]]]]]]]]<.[-]>>]
True	1893	80336	96	Counter({'+': 1727, '>': 113, '<': 21, '[': 9, ']': 9, '-': 7, '.': 7})	>>>>++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++>++++++++++++++++++>++++++++++++++++++>+>+>+>+>+>+>+>+++++++++++++++++++++++++++++++++++++++++++++++++>+++>++++++++++++++++++++>+>+>+>++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++>+++>++++++++++++++++++++>+>+>++++++++++++++++++++>+>+>+>++++++++++++++++++>++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++>+>++++++++++++++++++++>+++>++++>++++>++++>++++>++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++>+++>++++++++++++++++++>++++++++++++++++++>++++>++++++++++++++++++>+>++++++++++++++++++++>++++++++++++++++++++>++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++>++++>+++++++++++++++++++++++++++++++++++++++++++++++++>+++>++++++++++++++++++++>+>++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++>+++>++++++++++++++++++>+>++++++++++++++++++++>++++++++++++++++++++>+>++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++>++++++++++++++++++++>++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++>++++++++++++++++++>++++++++++++++++++>++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++>++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++>+++>++++>++++++++++++++++++++>+++++++++++++++++++++++++++++++++++++++++++++++++++><[<]<<<+++++++[->+++<]>[->++>+++<<]>+>-....>[[-<<.<+>>>]<.[->+<]<[-<+>>+<]>>>]<<<<[<]>[-.>]
True	978	32929	305	Counter({'+': 577, '>': 321, '-': 29, '<': 21, '[': 13, ']': 13, '.': 4})	>>+++++++>>++>>++++>>+++++++>>+>>++++>>+>>+++>>+>>+++++>>+>>++>>+>>++++++>>++>>++++>>+++++++>>+>>+++++>>++>>+>>+>>++++>>+++++++>>+>>+++++>>+>>+>>+>>++++>>+++++++>>+>>+++++>>++++++++++++++>>+>>+>>++++>>+++++++>>+>>+++++>>++>>+>>+>>++++>>+++++++>>+>>+++++>>+++++++++++++++++++++++++++++>>+>>+>>++++>>+++++++>>+>>+++++>>++>>+>>+>>+++++>>+>>++++++>>+>>++>>+>>++++++>>+>>++>>+>>++++++>>+>>++>>+>>++++++>>+>>++>>+>>++++++>>+>>++>>+>>++++++>>+>>++>>+>>++++++>>++>>++++>>+++++++>>+>>+++++>>+++++++>>+>>+++++>>+>>+>>+>>++++>>+>>++>>+>>++++++>>+>>+++++>>+++++++>>+>>++++>>+>>+>>++>>+++++>>+>>+++>>+>>++++>>+>>++>>+>>++++++>>+>>+++++>>+++++++++++++++++++>>++>>++>>+++>>++>>+>>++>>++++>>+++++++>>++>>+++++>>++++++++++>>+>>++>>++++>>+>>++>>+>>++++++>>++++++>>+>>+>>+++++>>+>>++++++>>++>>+++++>>+++++++>>++>>++++>>+>>++++++[<<]>>[>++++++[-<<++++++++++>>]<<++..------------------->[-<.>>+<]>[-<+>]>]<<[-[-[-[-[-[-[>++>]<+++++++++++++++++++++++++++++>]<++>]<++++++++++++++>]<+>]<++>]<<[->.<]<<]
True	822	188024	296	Counter({'+': 404, '>': 319, '-': 34, '<': 29, '[': 16, ']': 16, '.': 4})	->+>+++>>+>++>+>+++>>+>++>>>+>+>+>++>+>>>>+++>+>>++>+>+++>>++>++>>+>>+>++>++>+>>>>+++>+>>>>++>++>>>>+>>++>+>+++>>>++>>++++++>>+>>++>+>>>>+++>>+++++>>+>+++>>>++>>++>>+>>++>+>+++>>>++>>+++++++++++++>>+>>++>+>+++>+>+++>>>++>>++++>>+>>++>+>>>>+++>>+++++>>>>++>>>>+>+>++>>+++>+>>>>+++>+>>>>+++>+>>>>+++>>++>++>+>+++>+>++>++>>>>>>++>+>+++>>>>>+++>>>++>+>+++>+>+>++>>>>>>++>>>+>>>++>+>>>>+++>+>>>+>>++>+>++++++++++++++++++>>>>+>+>>>+>>++>+>+++>>>++>>++++++++>>+>>++>+>>>>+++>>++++++>>>+>++>>+++>+>+>++>+>+++>>>>>+++>>>+>+>>++>+>+++>>>++>>++++++++>>+>>++>+>>>>+++>>++++>>+>+++>>>>>>++>+>+++>>+>++>>>>+>+>++>+>>>>+++>>+++>>>+[[->>+<<]<+]+++++[->+++++++++<]>.[+]>>[<<+++++++[->+++++++++<]>-.------------------->-[-<.<+>>]<[+]<+>>>]<<<[-[-[-[>>+<++++++[->+++++<]]>++++++++++++++<]>+++<]++++++[->+++++++<]>+<<<-[->>>++<<<]>[->>.<<]<<]
True	410	859733	823	Counter({'+': 192, '>': 184, '<': 11, '[': 9, ']': 9, '-': 4, '.': 1})	->++>+++>+>+>++>>+>+>+++>>+>+>++>+++>+++>+>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>+>+>++>>>+++>>>>>+++>+>>>>>>>>>>>>>>>>>>>>>>+++>>>>>>>++>+++>+++>+>>+++>>>+++>+>+++>+>++>+++>>>+>+>+>+>++>+++>+>+>>+++>>>>>>>+>+>>>+>+>++>+++>+++>+>>+++>+++>+>+++>+>++>+++>++>>+>+>++>+++>+>+>>+++>>>+++>+>>>++>+++>+++>+>>+++>>>+++>+>+++>+>>+++>>+++>>+[[>>+[>]+>+[<]<-]>>[>]<+<+++[<]<<+]>>+[>]+++[++++++++++>++[-<++++++++++++++++>]<.<-<]
True	404	834577	811	Counter({'+': 191, '>': 181, '<': 10, '[': 9, ']': 9, '-': 3, '.': 1})	->++>+++>+>+>+++>>>>>>>>>>>>>>>>>>>>>>+>+>++>+++>++>>+++>+>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>+>+>>+++>>>>+++>>>+++>+>>>>>>>++>+++>+++>+>>+++>+++>+>+++>+>+++>+>++>+++>>>+>+>+>+>++>+++>+>+>>+++>>>>>>>+>+>>>+>+>++>+++>+++>+>>+++>+++>+>+++>+>++>+++>++>>+>+>++>+++>+>+>>+++>>>+++>+>>>++>+++>+++>+>>+++>>>+++>+>+++>+>>+++>>+++>>+[[>>+[>]+>+[<]<-]>>[>]<+<+++[<]<<+]>>>[>]+++>+[+[<++++++++++++++++>-]<++++++++++.<]
True	400	819524	803	Counter({'+': 188, '>': 180, '<': 10, '[': 9, ']': 9, '-': 3, '.': 1})	->++>+++>+>+>+++>>>>>>>>>>>>>>>>>>>>>>+>+>++>+++>++>>+++>+>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>+>+>>+++>>>>+++>>>+++>+>>>>>>>++>+++>+++>+>+++>+>>+++>>>+++>+>++>+++>>>+>+>+>+>++>+++>+>+>>+++>>>>>>>+>+>>>+>+>++>+++>+++>+>>+++>+++>+>+++>+>++>+++>++>>+>+>++>+++>+>+>>+++>>>+++>+>>>++>+++>+++>+>>+++>>>+++>+>+++>+>>+++>>+++>>+[[>>+[>]+>+[<]<-]>>[>]<+<+++[<]<<+]>+[>>]+++>+[+[<++++++++++++++++>-]<++++++++++.<]

By brute-force search I can now rule out all possible combinations of +-[]<>. of length 8 or less. So the true answer is anywhere between 8 and 400.

Open question: What is the relationship between length, steps and cells/memory used in (bf) quines?

Things that can be said about potential shorter quines (if they exist)

Symbol	ASCII value
+	43
-	45
.	46
<	60
>	62
[	91
]	93

since the quine cannot be of length zero, it must contain at least one `.`
-> it must contain at least one `+` since the symbol . has to be output
-> this alone places a naive lower step bound of 48 (43 additions, 1 output, 3 additions, 1 output) for outputting the two symbols only. Since this cannot be accomplished with two symbols alone, the lower bound increases depending on whether a loop is used or not.

Do quines fundamentally HAVE TO use loops? In some languages, which can inspect their own source, perhaps not (?). But in BF, it's impossible to output the same number of characters as the source length without using one.

Therefore, a BF quine has to contain at least +.[], meaning at some point all the corresponding ASCII values have to reside in memory, the instruction pointer points to them at some step and the instruction . is used to output them.

Things I've tried

Evolve them using genetic algorithms: https://github.com/void4/Brainfuck-Evolved (kurtjd/brainfuck-evolved#1)