# A formula on the weight distribution of linear codes with applications to AMDS codes

@article{Meneghetti2022AFO, title={A formula on the weight distribution of linear codes with applications to AMDS codes}, author={Alessio Meneghetti and Marco Antonio Pellegrini and Massimiliano Sala}, journal={ArXiv}, year={2022}, volume={abs/2003.14063} }

The determination of the weight distribution of linear codes has been a fascinating problem since the very beginning of coding theory. There has been a lot of research on weight enumerators of special cases, such as self-dual codes and codes with small Singleton's defect. We propose a new set of linear relations that must be satisfied by the coefficients of the weight distribution. From these relations we are able to derive known identities (in an easier way) for interesting cases, such as… Expand

#### References

SHOWING 1-10 OF 45 REFERENCES

On the Classification and Enumeration of Self-Dual Codes

- Computer Science, Mathematics
- J. Comb. Theory, Ser. A
- 1975

A complete classification is given of all [22, 11] and [24, 12] binary self-dual codes, giving the order of its group, the number of codes equivalent to it, and its weight distribution. Expand

An Upper Bound for Self-Dual Codes

- Mathematics, Computer Science
- Inf. Control.
- 1973

The general form that the weight distribution of a self-dual code over GF (2) and GF (3) can have is described and an explicit formula for this weight distribution when the minimum distance between codewords is made as large as possible is given. Expand

Weight distribution of Hermitian codes and matrices rank

- Computer Science, Mathematics
- Finite Fields Their Appl.
- 2019

This work shows that the computation of a given element of the weight distribution is equivalent to a less laborious task, which is the classification of some submatrices of the parity-check matrix according to their rank. Expand

Weight distributions of geometric Goppa codes

- Mathematics
- 1999

The in general hard problem of computing weight distributions of linear codes is considered for the special class of algebraic-geometric codes, defined by Goppa in the early eighties. Known results… Expand

A note on formally self-dual even codes of length divisible by 8

- Computer Science, Mathematics
- Finite Fields Their Appl.
- 2007

It is shown that any formally self-dual even binary code C of length n not divisible by 8 is balanced and the weight distribution of a balanced near-extremal f.s.d. even code of length a multiple of 8 is unique. Expand

On near-MDS codes

- Computer Science, Mathematics
- 1994

A family of codes obtained by weakening the restrictions in the definition of classical maximum-distance-separable (MDS) codes, which the authors call near-MDS (NMDS), which contains remarkable representatives such as the ternary Golay codes and extended quadratic-residue codes. Expand

Weight distributions of cyclic self-dual codes

- Mathematics, Computer Science
- IEEE Trans. Inf. Theory
- 2003

We give a 1-level squaring construction for binary repeated-root cyclic codes of length n=2/sup a/b, a/spl ges/1, b odd. This allows us to obtain the weight distributions of all cyclic binary… Expand

WEIGHT POLYNOMIALS OF SELF-DUAL CODES AND THE MacWILLIAMS IDENTITIES

- Mathematics
- 1970

Many error correcting codes are known to be self-dual. Hence the MacWilliams identities put a considerable restriction on the possible weight distribution of such a code. We show that this… Expand

Codes of Small Defect

- Mathematics, Computer Science
- Des. Codes Cryptogr.
- 1997

A condition on the minimum distance of a code to guarantee that the orthogonal code is an almost MDS code is presented, and Evaluation of the MacWilliams identities leads to a closed formula for the weight distribution which turns out to be completely determined for almost M DS codes up to one parameter. Expand

On the small-weight codewords of some Hermitian codes

- Computer Science, Mathematics
- J. Symb. Comput.
- 2016

For any affine-variety code we show how to construct an ideal whose solutions correspond to codewords with any assigned weight. We are able to obtain geometric characterizations for small-weight… Expand