On Optimal Binary One-Error-Correcting Codes of Lengths $2^{m}-4$ and $2^{m}-3$

Krotov, Denis S.; Östergård, Patric R. J.; Pottonen, Olli

doi:10.1109/tit.2011.2147758

Cited by 10 publications

(16 citation statements)

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“…This embedding result makes some facts known for centered functions (see, e.g., [1]) applicable for studying (n, 3) ′′′ op codes (for similar embedding result for (n, 3) ′′ op codes, see [4,Section 4]). It is worth to mention here another important common property of the considered classes of codes, which also can be derived from the results above, but actually has a more direct prove, found in [6].…”

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confidence: 58%

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On the binary codes with parameters of triply-shortened 1-perfect codes

Krotov¹

2011

Des. Codes Cryptogr.

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We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary $(n=2^m-3, 2^{n-m-1}, 4)$ code $C$, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the $n$-cube into six cells. An arbitrary binary $(n=2^m-4, 2^{n-m}, 3)$ code $D$, i.e., a code with parameters of a triply-shortened Hamming code, is a cell of an equitable family (but not a partition) from six cells. As a corollary, the codes $C$ and $D$ are completely semiregular; i.e., the weight distribution of such a code depends only on the minimal and maximal codeword weights and the code parameters. Moreover, if $D$ is self-complementary, then it is completely regular. As an intermediate result, we prove, in terms of distance distributions, a general criterion for a partition of the vertices of a graph (from rather general class of graphs, including the distance-regular graphs) to be equitable. Keywords: 1-perfect code; triply-shortened 1-perfect code; equitable partition; perfect coloring; weight distribution; distance distributionComment: 12 page

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confidence: 58%

“…The situation with (n, 3) ′′ op is different. There are such codes that cannot be represented as doubly-shortened 1-perfect [7,6]. Nevertheless, every (n, 3) ′′ op code is a cell of an equitable partition with quotient matrix ((0, 1, n−1, 0)(1, 0, n−1, 0)(1, 1, n−4, 2)(0, 0, n−1, 1)) [4].…”

Section: Introductionmentioning

confidence: 99%

On the binary codes with parameters of triply-shortened 1-perfect codes

Krotov¹

2011

Des. Codes Cryptogr.

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“…For n ≤ 15, we currently know not only the size of optimal codes, but we know all optimal codes up to symmetry; see [10,Sect. 7.1.4], [12], and [20]. Computational techniques have played a central role in obtaining those results, but we are now approaching the limits for such work.…”

Section: Introductionmentioning

confidence: 99%

The sextuply shortened binary Golay code is optimal

Östergård

2018

Des. Codes Cryptogr.

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The maximum size of unrestricted binary three-error-correcting codes has been known up to the length of the binary Golay code, with two exceptions. Specifically, denoting the maximum size of an unrestricted binary code of length n and minimum distance d by A(n, d), it has been known that 64 ≤ A(18, 8) ≤ 68 and 128 ≤ A(19, 8) ≤ 131. In the current computer-aided study, it is shown that A(18, 8) = 64 and A(19, 8) = 128, so an optimal code is obtained even after shortening the extended binary Golay code six times.

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“…After that work was completed, it was natural to think about the classification of perfect codes over larger alphabets, in particular,

{(q + 1, q^{q - 1}, 3)}_{q}

MDS codes. The current work is one step in that direction, along the ideas of , which presents an alternative approach for classifying the (15, 2 11 , 3) 2 codes.…”

Section: Introductionmentioning

confidence: 99%

“…After that work was completed, it was natural to think about the classification of perfect codes over larger alphabets, in particular, (q + 1, q q−1 , 3) q MDS codes. The current work is one step in that direction, along the ideas of [15], which presents an alternative approach for classifying the (15, 2 11 , 3) 2 codes.The following is known about short optimal (n, q n−2 , 3) q MDS codes, which have the parameters of subcodes of (q + 1, q q−1 , 3) q perfect codes. The (3, q, 3) q codes are unique, and the (4, q 2 , 3) q codes correspond to pairs of mutually orthogonal Latin squares of order q, which have been classified for q ≤ 8 (these will be further discussed in Section 2.1).…”

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confidence: 99%