The Classification of Some Perfect Codes

Avgustinovich, S. V.; Heden, Olof; Solov’eva, Faina I.

doi:10.1023/b:desi.0000015891.01562.c1

Cited by 25 publications

(35 citation statements)

References 7 publications

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“…2, we get that the word01 ∈ D ⊥ . Also, by Corollary 4.1, we get that there exist some wordsd (0) ,d (1) , . .…”

Section: Theorem 6 2 the Only Integers P For Which There Exists A Mamentioning

confidence: 83%

“…By [1], if C is a Hamming code of length n = 2 m − 1 and V(C, λ) is nonlinear, then dim( V(C, λ) ) = 2n + 1 − m. Hence, we may conclude that…”

Section: Theorem 4 4 (Vasil'ev) For Any Perfect Code C Of Length N Amentioning

confidence: 92%

“…, I * t . The lemma and proposition below are straightforward consequences of the results of [1] or [2].…”

Section: Simplex Codes and The Associated Extended Fundamental Partitmentioning

confidence: 99%

“…Let us denote this Hamming code by D . From (13) we get that there are some words y (0) ,ȳ (1) , . .…”

Section: Theorem 6 2 the Only Integers P For Which There Exists A Mamentioning

confidence: 99%

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Maximal partial packings of $${Z_2^n}$$ with perfect codes

Westerbäck

2007

Des Codes Crypt

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A maximal partial Hamming packing of Z n 2 is a family S of mutually disjoint translates of Hamming codes of length n, such that any translate of any Hamming code of length n intersects at least one of the translates of Hamming codes in S. The number of translates of Hamming codes in S is the packing number, and a partial Hamming packing is strictly partial if the family S does not constitute a partition of Z n 2 . A simple and useful condition describing when two translates of Hamming codes are disjoint or not disjoint is proved. This condition depends on the dual codes of the corresponding Hamming codes. Partly, by using this condition, it is shown that the packing number p, for any maximal strictly partial Hamming packing of Z n 2 , n = 2 m −1, satisfies m + 1 ≤ p ≤ n − 1.It is also proved that for any n equal to 2 m − 1, m ≥ 4, there exist maximal strictly partial Hamming packings of Z n 2 with packing numbers n − 10, n − 9, n − 8, . . . , n − 1. This implies that the upper bound is tight for any n = 2 m − 1, m ≥ 4.All packing numbers for maximal strictly partial Hamming packings of Z n 2 , n = 7 and 15, are found by a computer search. In the case n = 7 the packing number is 5, and in the case n = 15 the possible packing numbers are 5, 6, 7, . . . , 13 and 14.

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“…2, we get that the word01 ∈ D ⊥ . Also, by Corollary 4.1, we get that there exist some wordsd (0) ,d (1) , . .…”

Section: Theorem 6 2 the Only Integers P For Which There Exists A Mamentioning

confidence: 83%

“…By [1], if C is a Hamming code of length n = 2 m − 1 and V(C, λ) is nonlinear, then dim( V(C, λ) ) = 2n + 1 − m. Hence, we may conclude that…”

Section: Theorem 4 4 (Vasil'ev) For Any Perfect Code C Of Length N Amentioning

confidence: 92%

“…, I * t . The lemma and proposition below are straightforward consequences of the results of [1] or [2].…”

Section: Simplex Codes and The Associated Extended Fundamental Partitmentioning

confidence: 99%

“…Let us denote this Hamming code by D . From (13) we get that there are some words y (0) ,ȳ (1) , . .…”

Section: Theorem 6 2 the Only Integers P For Which There Exists A Mamentioning

confidence: 99%

See 2 more Smart Citations

Maximal partial packings of $${Z_2^n}$$ with perfect codes

Westerbäck

2007

Des Codes Crypt

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“…In the sequel, we only consider codes of rank greater by one than the rank of the Hamming code of the same length. One can prove (see also [15]) that an arbitrary perfect code of length 2n + 1 and of rank (2n + 1) − log(2n + 2) + 1 = 2n + 1 − log(n + 1) (i.e., of rank greater by one than that of the Hamming code of length 2n + 1) is a Vasil'ev code obtained from the Hamming code H of length n. In our notations, this code is of the form…”

Section: Corollarymentioning

confidence: 97%

On the Structure of Symmetry Groups of Vasil’ev Codes

2005

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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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The Classification of Some Perfect Codes

Cited by 25 publications

References 7 publications

Maximal partial packings of $${Z_2^n}$$ with perfect codes

Maximal partial packings of $${Z_2^n}$$ with perfect codes

On the Structure of Symmetry Groups of Vasil’ev Codes

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

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