Translation-invariant propelinear codes

Abstract. Let F n be the n-dimensional vector space over Z 2 . A (binary) 1-perfect partition of F n is a partition of F n into (binary) perfect single error-correcting codes or 1-perfect codes.We define two metric properties for 1-perfect partitions: uniformity and distance invariance. Then we prove the equivalence between these properties and algebraic properties of the code (the class containing the zero vector). In this way, we characterize 1-perfect partitions obtained using 1-perfect translation invariant and not translation invariant propelinear codes.The search for examples of 1-perfect uniform but not distance invariant partitions enabled us to deduce a non-Abelian propelinear group structure for any Hamming code of length greater than 7.

show abstract

“…As can be seen in [9], any translation invariant propelinear code can be viewed as a group isomorphic to a subgroup of Z…”

Section: -Perfect Uniform Partitionsmentioning

confidence: 99%

“…In G 7 , any π w (w ∈ G 7 ) is a product of disjoint transpositions (see [9] or [2]). Then, π c and π x are also a product of disjoint transpositions.…”

Section: -Perfect Uniform Not Distance Invariant Partitionsmentioning

confidence: 99%

1-Perfect Uniform and Distance Invariant Partitions

Rifà

Pujol

Borges

2001

AAECC

Self Cite

View full text Add to dashboard Cite

show abstract

“…For example, in propelinear codes (see [8]), it would be more natural to use the partition on F n given by…”

Section: -Perfect Partitionsmentioning

confidence: 99%

“…For example, using uniform 1-perfect partitions, we can give the propelinear structure to all the classes of the partition (see [8]). …”

Section: Conclusion and Further Researchmentioning

confidence: 99%

“…The STS which results in this code is formed by the following triples (see [1]): (1, 2, 3), (1,4,5), (1,6,7), (1,8,9), (1, 10, 11), (1, 12, 13), (1,14,15), (2,4,6), (2,5,7), (2,8,10), (2, 9, 11), (2,12,14), (2,13,15), (3,4,7), (3,5,6), (3,8,12), (3, 9, 13), (3,10,14), (3,11,15), (4,8,15), (4,9,14), (4, 10, 13), (4, 11, 12), (5,8,11), (5, 9, 12), (5, 10, 15), (5,13,14), …”

mentioning

confidence: 99%

See 1 more Smart Citation

Well-Ordered Steiner Triple Systems and 1-Perfect Partitions of the n-cube

Rifà¹

1999

SIAM J. Discrete Math.

View full text Add to dashboard Cite

Abstract. Binary 1-perfect codes which give rise to partitions of the n-cube are presented. The 1-perfect partitions are characterized as homomorphic images of simple algebraic structures on F n and are constructed starting from a particular case of a structure defined in F n . A special property (so-called well-ordering) of ST S(n) is given in such a way that for this kind of STS it is possible to define the algebraic structure we need in F n and to construct 1-perfect partitions of the n-cube.These 1-perfect partitions give us a kind of 1-perfect code for which it is easy to do the coding and decoding. Furthermore, there exists a syndrome which allows us to perform error correction. We present systematic codes of length n = 15 and we give examples of how to do the coding, decoding, and error correction.

show abstract

Linear and cyclic codes over direct product of finite chain rings

Borges

Fernández-Córdoba

Ten-Valls

2017

Math Methods in App Sciences

View full text Add to dashboard Cite

We introduce a new type of linear and cyclic codes. These codes are defined over a direct product of two finite chain rings. The definition of these codes as certain submodules of the direct product of copies of these rings is given and the cyclic property is defined. Cyclic codes can be seen as submodules of the direct product of polynomial rings. Generator matrices for linear codes and generator polynomials for cyclic codes are determined.

show abstract

Translation-invariant propelinear codes

Cited by 76 publications

References 9 publications

1-Perfect Uniform and Distance Invariant Partitions

1-Perfect Uniform and Distance Invariant Partitions

Well-Ordered Steiner Triple Systems and 1-Perfect Partitions of the n-cube

Linear and cyclic codes over direct product of finite chain rings

Contact Info

Product

Resources

About