Two optimal one-error-correcting codes of length 13 that are not doubly shortened perfect codes

Östergård, Patric R. J.; Pottonen, Olli

doi:10.1007/s10623-010-9450-4

Cited by 9 publications

(19 citation statements)

References 13 publications

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“…All these results still leave the classification problem open for lengths 12 and 13. It is known [5] that not all such codes can be obtained by shortening codes of length 14 or 15.…”

Section: A Survey Of Old Resultsmentioning

confidence: 99%

“…1 14179 64 8511 2048 39 2 45267 96 90 3072 3 3 41 128 3114 4096 9 4 66449 192 55 6144 4 6 137 256 1247 8192 1 8 44529 384 39 12288 4 12 159 512 403 16384 1 16 32193 768 35 24576 1 24 89 1024 82 73728 1 32 20813 1152 1 147456 1 48 98 1536 15 the unique (13,256,4) code that cannot be lengthened to a (14,512,4) code has an automorphism group of order 384. It turns out that one detail in [5] is incorrect: shortening the (two) (13, 512, 3) codes that cannot be lengthened to (15,2048,3) codes always leads to (12,256,3) codes that cannot be lengthened to (15,2048,3) codes.…”

Section: Properties Of the Classified Codesmentioning

confidence: 99%

“…In [5] two equivalence classes of (13, 512, 3) codes that cannot be lengthened were found, in addition to the 117819 equivalence classes that can be lengthened. Our results show that the two exceptional codes found in [5] are the only ones with this property. Moreover, they have equivalent extensions, so there is a unique (14,512,4) code that cannot be lengthened to a (16,2048,4) code; the automorphism group of this code has order 768.…”

mentioning

confidence: 95%

“…In general, a complete characterization or classification of such codes does not seem feasible, but the classification problem can be addressed for small parameters and general properties of these codes can be studied. For example, the issue whether codes with these parameters can be lengthened to perfect codes has attracted some interest in the literature [3], [4], [5], [6]. For i = 1, every code (2) can be lengthened to a perfect code and this can be done in a unique way up to equivalence [3].…”

Section: Introductionmentioning

confidence: 99%

“…1 782 64 15534 3072 15 2 4464 96 48 4096 59 3 55 128 6988 6144 5 4 11412 192 51 8192 13 6 71 256 3245 12288 3 8 19902 384 16 16384 7 12 37 512 1391 24576 1 16 27406 768 19 32768 1 24 54 1024 475 49152 1 32 25506 1536 26 98304 1 48 73 2048 162 It is known [5] that not all (12, 256, 3) and (13, 512, 3) codes can be lengthened to (15,2048,3) codes (and analogously for the extended codes with d = 4). In [5] two equivalence classes of (13, 512, 3) codes that cannot be lengthened were found, in addition to the 117819 equivalence classes that can be lengthened. Our results show that the two exceptional codes found in [5] are the only ones with this property.…”

mentioning

confidence: 99%

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

Krotov

Östergård

Pottonen

2011

IEEE Trans. Inform. Theory

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“…All these results still leave the classification problem open for lengths 12 and 13. It is known [5] that not all such codes can be obtained by shortening codes of length 14 or 15.…”

Section: A Survey Of Old Resultsmentioning

confidence: 99%

Section: Properties Of the Classified Codesmentioning

confidence: 99%

mentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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

Krotov

Östergård

Pottonen

2011

IEEE Trans. Inform. Theory

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

Krotov

2010

Des. Codes Cryptogr.

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We show that any binary (n = 2 m − 3, 2 n−m , 3) code C1 is a part of an equitable partition (perfect coloring) {C1, C2, C3, C4} of the n-cube with the parameters ((0, 1, n − 1, 0)(1, 0, n − 1, 0)(1, 1, n − 4, 2)(0, 0, n − 1, 1)). Now the possibility to lengthen the code C1 to a 1-perfect code of length n+2 is equivalent to the possibility to split the part C4 into two distance-3 codes or, equivalently, to the biparticity of the graph of distances 1 and 2 of C4. In any case, C1 is uniquely embeddable in a twofold 1-perfect code of length n + 2 with some structural restrictions, where by a twofold 1-perfect code we mean that any vertex of the space is within radius 1 from exactly two codewords.The hypercube H n = (V (H n ), E(H n )) of dimension n is the graph whose vertices are the all binary n-words, two words being adjacent if and only if they differ in exactly one position.d(·, ·) -the Hamming distance, i.e., the natural graph distance in H n . 0 = 0 . . . 0 (the all-zero word),1 = 1 . . . 1 (the all-one word).A binary code C of length n and code (or minimal) distance d, or (n, |C|, d) code, is a subset of V (H n ) such that d(x,ȳ) ≥ d for any differentx andȳ from C.A partition {C 1 , . . . , C r } of V (H n ) into r nonempty parts is said to be equitable with parameters (S ij ) n i,j=1 if for every i, j ∈ {1, . . . , r} every vertexx from C i has exactly S ij neighbors from C j (the corresponding r-valued function on V (H n ) is known as a perfect coloring).A binary code C ⊂ V (H n ) is said to be 1-perfect if every vertexx ∈ V (H n ) is at the distance 0 or 1 from exactly one codeword. Equivalently, {C, V (H n ) \ C} is an equitable partition with parameters ((0, n)(1, n − 1)). Equivalently, C is a (2 m − 1, 2 2 m −m−1 , 3) code, n = 2 m − 1.We will say that a multiset B ⊂ V (H n ) is a twofold 1-perfect code if every vertexx ∈ V (H n ) is at the distance 0 or 1 from exactly two codewords of B. We will say that a multiset B ⊂ V (H n ) is splittable if it can be represented as the (multiset) union of two distance-3 codes; otherwise B is unsplittable. The existence of unsplittable twofold 1-perfect codes was proved in [6].We say that a code C ′ if obtained by shortening from a code C ⊂ V (H n ) if C ′ = {x ∈ V (H n−1 ) |x0 ∈ C}. Respectively, C ′′ is doubly-shortened from C if 1

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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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Two optimal one-error-correcting codes of length 13 that are not doubly shortened perfect codes

Cited by 9 publications

References 13 publications

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

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

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

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

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