A075930 -id:A075930 - cp's OEIS frontend

A178374 a(n) = A075930(n)^A178307(n).

343, 1331, 169, 2744, 6859, 441, 10648, 390625, 676, 21952, 29791, 1500625, 1369, 54872, 68921, 1764, 85184, 2209, 5764801, 6250000, 2704, 166375, 175616, 714924299, 3721, 238328

Offset: 1

Jeremy Gardiner, May 26 2010

a(n) is in A001969 for all n.

			A075930(1)=7, A178307(1)=3, a(1)=7^3=343.

Cf. A001969, A075930, A178307, A178373.

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

Original entry on oeis.org

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

Bob Jenkins (bob_jenkins(AT)burtleburtle.net)

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.

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

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

A194851 is a subsequence.

A075926 List of codewords in binary lexicode with Hamming distance 3 written as decimal numbers.

Original entry on oeis.org

0, 7, 25, 30, 42, 45, 51, 52, 75, 76, 82, 85, 97, 102, 120, 127, 385, 390, 408, 415, 427, 428, 434, 437, 458, 461, 467, 468, 480, 487, 505, 510, 642, 645, 667, 668, 680, 687, 689, 694, 713, 718, 720, 727, 739, 740, 762, 765, 771, 772, 794, 797, 809

Offset: 0

Bob Jenkins (bob_jenkins(AT)burtleburtle.net)

This is also the (infinite) d=3 Hamming code.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
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.
Ari Trachtenberg, Error-Correcting Codes on Graphs: Lexicodes, Trellises and Factor Graphs

Cf. A075928, A075929, A075930, A075927, A057716, A075931.

A075931 List of codewords in binary lexicode with Hamming distance 5 written as decimal numbers.

Original entry on oeis.org

0, 31, 227, 252, 805, 826, 966, 985, 1354, 1365, 1449, 1462, 1647, 1648, 1676, 1683, 6182, 6201, 6341, 6362, 6915, 6940, 7136, 7167, 7532, 7539, 7567, 7568, 7753, 7766, 7850, 7861, 10315, 10324, 10408, 10423, 11118, 11121, 11149, 11154

Offset: 0

Bob Jenkins (bob_jenkins(AT)burtleburtle.net)

Rémy Sigrist, Table of n, a(n) for n = 0..8191
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.
Bob Jenkins, Tables of Binary Lexicodes
Ari Trachtenberg, Error-Correcting Codes on Graphs: Lexicodes, Trellises and Factor Graphs

Cf. A075928, A075929, A075930, A075932, A075933, A075924, A075926, A075937.

a=vector(40); n=0; for (k=0, 11154, if (n==0 || vecmin(apply(o -> hammingweight(bitxor(k, o)), a[1..n]))>=5, print1 (a[n++]=k", "))) \\ Rémy Sigrist, Feb 09 2021

A075934 List of codewords in binary lexicode with Hamming distance 6 written as decimal numbers.

Original entry on oeis.org

0, 63, 455, 504, 1611, 1652, 1932, 1971, 2709, 2730, 2898, 2925, 3294, 3297, 3353, 3366, 12365, 12402, 12682, 12725, 13830, 13881, 14273, 14334, 15064, 15079, 15135, 15136, 15507, 15532, 15700, 15723, 20630, 20649, 20817, 20846, 22237

Offset: 0

Bob Jenkins (bob_jenkins(AT)burtleburtle.net)

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.
Bob Jenkins, Tables of Binary Lexicodes
Ari Trachtenberg, Error-Correcting Codes on Graphs: Lexicodes, Trellises and Factor Graphs

Cf. A075928, A075929, A075930, A075935, A075936, A075931, A075928, A075940.

A075937 List of codewords in binary lexicode with Hamming distance 7 written as decimal numbers.

Original entry on oeis.org

0, 127, 1927, 2040, 6553, 6630, 7710, 7777, 10922, 10965, 11565, 11602, 13107, 13132, 13492, 13515, 19252, 19275, 19635, 19660, 21165, 21202, 21802, 21845, 24990, 25057, 26137, 26214, 30727, 30840, 32640, 32767, 98731, 98772, 99884

Offset: 0

Bob Jenkins (bob_jenkins(AT)burtleburtle.net)

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.

Rémy Sigrist, Table of n, a(n) for n = 0..8191
Bob Jenkins, Tables of Binary Lexicodes
Rémy Sigrist, C program for A075937
Ari Trachtenberg, Error-Correcting Codes on Graphs: Lexicodes, Trellises and Factor Graphs

Cf. A075928, A075929, A075930, A075938, A075939, A075940, A075931, A075943.

C
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See Links section.
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A075944 Basis for code in A075943.

Original entry on oeis.org

511, 15887, 116275, 412245, 693418, 3165290, 12603032, 21139155, 38803550, 201349286, 336626733, 609260174, 3221261509, 5374090380, 25769843817, 43016936625, 78384372260, 412317013686, 687266122367, 1238028666383

Offset: 0

Bob Jenkins (bob_jenkins(AT)burtleburtle.net)

nonn

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.

Bob Jenkins, Tables of Binary Lexicodes
Ari Trachtenberg, Error-Correcting Codes on Graphs: Lexicodes, Trellises and Factor Graphs

Cf. A075928, A075929, A075930, A075943, A075945, A075947, A075938, A075950.

a(n) = b(2^n), where b is A075943, the binary lexicode, d=9.

b(n) = XOR(a(i)) for all i where the i-th bit is set in n.

A075945 Positions of check bits in code in A075943.

Original entry on oeis.org

255, 3855, 13107, 21845, 27306, 38506, 71576, 92115, 101470, 138406, 172589, 240014, 272069, 380556, 536169, 679601, 847140, 1072054, 1258879, 1397519, 2005450, 2121415, 2496892, 2738813, 3993667, 4241896, 4617503, 5821704

Offset: 0

Bob Jenkins (bob_jenkins(AT)burtleburtle.net)

nonn

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.

Bob Jenkins, Tables of Binary Lexicodes
Ari Trachtenberg, Error-Correcting Codes on Graphs: Lexicodes, Trellises and Factor Graphs

Cf. A075928, A075929, A075930, A075943, A075944, A075948, A075939, A075951.

A075946 List of codewords in binary lexicode with Hamming distance 10 written as decimal numbers.

Original entry on oeis.org

0, 1023, 31775, 32736, 232551, 233368, 258168, 258951, 824491, 825172, 846004, 846667, 989388, 990003, 1008851, 1009452, 1386837, 1387178, 1398090, 1398453, 1484082, 1484493, 1497389, 1497810, 1687038, 1687041, 1688033

Offset: 0

Bob Jenkins (bob_jenkins(AT)burtleburtle.net)

nonn

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.

Bob Jenkins, Tables of Binary Lexicodes
Ari Trachtenberg, Error-Correcting Codes on Graphs: Lexicodes, Trellises and Factor Graphs

Cf. A075928, A075929, A075930, A075947, A075948, A075943, A075940, A075952.

A075949 List of codewords in binary lexicode with Hamming distance 11 written as decimal numbers.

Original entry on oeis.org

0, 2047, 129055, 131040, 932067, 933660, 1032444, 1033987, 3299692, 3301011, 3383667, 3384972, 3957135, 3958384, 4037008, 4038255, 5548725, 5549386, 5591722, 5592405, 5935702, 5936553, 5990985, 5991862, 6747097, 6747174

Offset: 0

Bob Jenkins (bob_jenkins(AT)burtleburtle.net)

nonn

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.

Bob Jenkins, Tables of Binary Lexicodes
Ari Trachtenberg, Error-Correcting Codes on Graphs: Lexicodes, Trellises and Factor Graphs

Cf. A075928, A075929, A075930, A075950, A075951, A075952, A075943, A075955.

cp's OEIS Frontend

A178374 a(n) = A075930(n)^A178307(n).

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

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A075926 List of codewords in binary lexicode with Hamming distance 3 written as decimal numbers.

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A075931 List of codewords in binary lexicode with Hamming distance 5 written as decimal numbers.

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PARI

A075934 List of codewords in binary lexicode with Hamming distance 6 written as decimal numbers.

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A075937 List of codewords in binary lexicode with Hamming distance 7 written as decimal numbers.

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C

A075944 Basis for code in A075943.

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A075945 Positions of check bits in code in A075943.

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A075946 List of codewords in binary lexicode with Hamming distance 10 written as decimal numbers.

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A075949 List of codewords in binary lexicode with Hamming distance 11 written as decimal numbers.

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