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**Note sur la notion d’équivalence entre deux codes linéaires.**
*(French)*
Zbl 0632.94009

In this paper it is shown that the existence of a Hamming isometry sending a code C onto a code C’ implies the existence of a Hamming isometry sending C onto C’ whose restrictions to the simple components of C are semilinear. (A code is called simple if it is not the direct sum of two or more smaller codes having disjoint supports, the support of a code is the union of the supports of its elementary words (which are the words x in C such that there does not exist a word x’ in C with supp(x’) contained in supp(x)).)

In the case where C is a subspace of \({\mathbb{F}}^ n\), where \({\mathbb{F}}\) is of prime order, it has been proved that such a Hamming isometry is linear.

The paper is well written and I think the results are nice. However example (3) on page 181 is incorrect: The code \(C={\mathbb{F}}^ 2_ 2\) is cyclic but not simple since \(C=<(1\quad 0)>\oplus <(0\quad 1)>.\)

In the case where C is a subspace of \({\mathbb{F}}^ n\), where \({\mathbb{F}}\) is of prime order, it has been proved that such a Hamming isometry is linear.

The paper is well written and I think the results are nice. However example (3) on page 181 is incorrect: The code \(C={\mathbb{F}}^ 2_ 2\) is cyclic but not simple since \(C=<(1\quad 0)>\oplus <(0\quad 1)>.\)

Reviewer: H.J.Tiersma

### MSC:

94B05 | Linear codes (general theory) |

20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |

Full Text:
DOI

### References:

[1] | Bonneau, P., Codes et Combinatoire, Thèse de 3eme Cycle (1984), Paris VI |

[2] | Bogart, K.; Goldberg, D.; Gordon, J., An elementary proof of the MacWilliams theorem on equivalence of codes, Inform. Control, 37, 19-22 (1978) · Zbl 0382.94021 |

[3] | Goldberg, D., A generalized weight for linear codes and a Witt-MacWilliams theorem, J. Combin. Theory Ser. A, 29, 363-367 (1980) · Zbl 0456.94011 |

[4] | MacWilliams, F. J., Error-correcting codes for multiple level transmission, Bell System Tech. J., 40, 281-308 (1961) |

[5] | MacWilliams, F. G., Combinatorial problems of elementary groups, (Ph.D. Thesis (1962), Harvard University) |

[6] | MacWilliams, F. J.; Sloane, N. J.A, The Theory of Error Correcting Codes (1977), North-Holland: North-Holland Amsterdam · Zbl 0369.94008 |

[7] | Welsh, N. D.A, Matroid Theory (1976), Academic Press: Academic Press New York · Zbl 0343.05002 |

[8] | Aigner, M., Combinatorial Theory (1979), Springer: Springer Berlin · Zbl 0415.05001 |

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