TITLE: Generation of max Hamming distance alphabet (24,10)
QUESTION [0 upvotes]: A related question to Generating a binary code with maximized Hamming distance :
Any ideas of a good way to find a set of 255 code words of length 24 with Hamming distance 10? According to this overview table of A(n,d), it should be feasible. In other words, I'm trying to make a (preferably linear) code mapping 8 bit information to 24 bit code words.
I understand that (to the current state of knowledge) there is no definitive way of establishing such an alphabet, but I'm looking for ideas to (relatively effectively) traverse through some subset of the full set to obtain a good performing alphabet.

REPLY [4 votes]: You are misunderstanding the table. The entry $136-268$ means that $136$ is the best known lower bound for the largest code, and $268$ is the best known upper bound. Practically, this means that

The largest code that has ever been discovered has size $136$ (the only way to prove the largest code has size at least $s$, is to construct a code with $s$ vectors).

We have a proof that no code of size $269$ can be constructed, but there are no non-existence proofs for any smallest numbers.

This means that for any $s\in \{137,\cdots,268\}$, we have not yet been able to construct a code of size $s$, yet we cannot rule out the existence of a code of size $s$.


Therefore, you should not hope to find a code of size $255$. The best you can hope for, without putting in significant time and effort researching a new code, is the current best code of size $136$. I have provided this code below, in case it is large enough to be useful to you. I got this code from the arxiv article below. Rather, I used wrote some code to generate the code according to the method written in the paper.

Antti Laaksonen and Patric R. J. Östergård, Constructing Error-Correcting Binary Codes Using Transitive Permutation Groups, 2016, https://arxiv.org/abs/1604.06022

  1    100101010010000111111111
  2    001000101000100010011111
  3    010011100100110011101111
  4    000110101111000110101111
  5    001010110010101100100111
  6    010010111001011101101101
  7    000110101100111100111000
  8    100001010110010001000001
  9    001101000011110100011101
 10    001110110101110010000000
 11    011010010010110000011010
 12    001000101011011000001000
 13    111100111011111100111011
 14    011011000001011011000001
 15    001111011001110111101011
 16    000110010011011111000110
 17    101000010001111001100010
 18    011100110011000001100001
 19    001011100001000111111100
 20    010110001100101010000110
 21    100000000110111000111111
 22    101111101110110000100101
 23    111101010111110011101100
 24    111111110001000010010111
 25    010111100011101010111010
 26    001011011101000000001101
 27    100110011011100010011100
 28    101101001011001001000111
 29    011110010101101110111101
 30    100001101001100001101001
 31    010111011100010111011100
 32    101011010110101011010110
 33    101101110100100000111010
 34    110101110001011001011000
 35    101110111101011011111110
 36    110011010011101101001011
 37    100111000101000011001010
 38    110110110110001100011110
 39    101011101001101110000010
 40    010111011111011000110111
 41    101111010010011100110000
 42    011000110111010111011111
 43    010110001111110001001001
 44    101010000110010110001100
 45    010101111010110010000011
 46    001101000000011010101110
 47    110011101100010000010010
 48    110101001110001000101000
 49    100011110000111110011101
 50    101010111010000101011001
 51    011001011011000110010000
 52    111101010100111100000111
 53    110100100001111010100101
 54    100100111101010100010101
 55    110011010000100010100000
 56    100101101110111111001110
 57    111011110101110101110001
 58    110111101011010111100000
 59    100100000010100100000010
 60    011000110100011000110100
 61    010111100000000100001001
 62    000101100110000010010100
 63    100010100000001001000100
 64    010001111110100100111101
 65    100010011100100101101110
 66    110000011010111101010100
 67    111001100010101000010001
 68    000011001010100111000101
 69    011100110000101111010010
 70    111000111100111011001001
 71    110001001001010110001111
 72    000111111100001011100001
 73    111011001001111000111100
 74    111000111111100000000110
 75    101110000010101011101001
 76    000001011101111010011010
 77    001111100110011001011011
 78    101101110111001110001001
 79    010000010010001010001101
 80    010000101110001101000011
 81    101001001100001110110101
 82    000110010000110001110101
 83    000000110010110111101000
 84    000101011001101100100100
 85    110100011101100011110011
 86    100010100011110011010011
 87    011011111110011110101010
 88    000010010100000110010011
 89    100000111010001010110010
 90    010100101010010001111110
 91    001001000100101101001000
 92    011001011000001001111011
 93    001001111011111011110101
 94    011110101010011110010101
 95    001001000111100010100011
 96    011000001000110100100001
 97    000000001101010011100100
 98    010100010101010100101010
 99    001100011110110110110110
100    000111111111100101010010
101    010001000001100001010110
102    000001100001011100110011
103    111100111000000110101100
104    111100000100000001011101
105    110000100101100110011000
106    011111111010101001001100
107    110111101000101101110111
108    010101000110111111110001
109    001010001101111101010111
110    100010011111111110100001
111    011000001011101111101110
112    101100100100010111100011
113    100011110011010000101110
114    001110001001000000110010
115    000010010111001001111000
116    110001110101001111100110
117    000100001001001111011001
118    011111000110000101100110
119    101100100111101101110100
120    101101001000110011010000
121    111111001110100110011011
122    111100000111011010010010
123    111010100101001000101011
124    110011101111001011011101
125    011101101101010010111001
126    010011100111111100000100
127    110010000011000100110101
128    011010101110100011110000
129    001001111000010101000110
130    101001001111010101111010
131    111110100001110101001110
132    110010000000011111111010
133    111110011101001101000000
134    100110011000011000001011
135    111010011010010011100111
136    000100110101101001001111
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