A322074 Maximum number of codewords a binary self-dual code of length 4n can have with Hamming weight 2n (half of length).

2, 14, 32, 198, 512, 2972, 8192, 45638, 131072

Offset: 1

Author

Nathan J. Russell, Nov 25 2018

Keywords

• nonn
• more

Comments

All binary self-dual codes of length 2*18 = 36 have a(9) = 131072 or fewer codewords with Hamming weight 18. In fact, there is only one binary self-dual code of length 36 that has 131072 codewords with Hamming weight 18.

There is at least one binary self-dual code of length 4n having a(n) codewords of weight 2n. However, the code may not be unique. There are two binary self-dual codes of length 4*4=16 having a(4)=198 codewords with Hamming weight 2*4=8.

All binary self-dual codes must be even length and all codewords must have an even Hamming weight. Only codewords with a length that is a multiple of 4 can have codewords with a Hamming weight equal to half the length of the code.

Links

• W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, 2003, Page 7, 252-330, 338-393.

Crossrefs

Cf. A322073, A321969, A296086, A001405, A000984.