A075928 List of codewords in binary lexicode with Hamming distance 4 written as decimal numbers.

0, 15, 51, 60, 85, 90, 102, 105, 150, 153, 165, 170, 195, 204, 240, 255, 771, 780, 816, 831, 854, 857, 869, 874, 917, 922, 934, 937, 960, 975, 1011, 1020, 1285, 1290, 1334, 1337, 1360, 1375, 1379, 1388, 1427, 1436, 1440, 1455, 1478, 1481, 1525

Offset: 0

Author

Bob Jenkins (bob_jenkins(AT)burtleburtle.net)

Keywords

• nonn
• easy
• base

Comments

The lexicode of Hamming distance d is constructed greedily by stepping through the binary vectors in lexicographic order and accepting a vector if it is at Hamming distance at least d from all already-chosen vectors.

The code is linear and infinite.

This is also the (infinite) d=4 Hamming code.

Lexicodes with even Hamming distance can be constructed from the preceding lexicode of odd Hamming distance by prepending a single parity bit.

Links

• J. H. Conway and N. J. A. Sloane, Lexicographic codes: error-correcting codes from game theory, IEEE Transactions on Information Theory, 32:337-348, 1986.
• R. W. Hamming, Error Detecting and Error Correcting Codes, Bell System Tech. J., Vol. 29, April, 1950, pp. 147-160.
• Bob Jenkins, Tables of Binary Lexicodes
• Ari Trachtenberg, Error-Correcting Codes on Graphs: Lexicodes, Trellises and Factor Graphs

Crossrefs

Cf. A075929, A075930, A075926, A075934, A075944, A075945, A075946, A075937, A075949, etc.

A194851 is a subsequence.