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Isolated Complementary Binary Linear Block Codes
ICBLBCs are binary linear block codes composed of sub-codes that feature
isolation from each other.
A binary linear block code encodes a certain number of data bits into a
codeword of a certain length. A codeword of n bits uniquely represents a block
of k data bits. Binary linear block codes are often characterized by the
minimum Hamming distance d between any two codewords.
A sub-code is a code that consists of a subset of the codewords of a larger
code. Complementary codes are non-overlapping sub-codes of the same larger
code.
An ICBLBC is a set of two or more complementary codes that feature isolation
from each other. Isolation means that there is a minimum Hamming distance from
sub-code to sub-code. For isolation to be useful for unambiguous encapsulation,
it must be larger than the minimum Hamming distance within each sub-code.
This project seeks ICBLBCs consisting of only two sub-codes.
Definitions
The terms "Isolated Complementary Binary Linear Block Code", "ICBLBC",
"complementary", "sub-code", and "isolation" as used in this description are my
own. These terms are subject to change upon review of this description by
members of the coding theory community.
Example
The following ICBLBC contains a total of eight codewords divided into two
sub-codes of four codewords each:
- 00000, 00011, 00101, 01001
- 01110, 10110, 11010, 11100
Each codeword is five bits long (n = 5). Within each sub-code, each codeword
represents two data bits (k = 2). Considering the ICBLBC as a whole, each
codeword represents three data bits (k = 3).
The minimum Hamming distance within each sub-code is two. The minimum Hamming
distance from one sub-code to the other (the isolation) is three.