## 1 Introduction

Let be the Galois field with elements, . Let be the space of -dimensional vectors over . We denote by an -linear code of length , dimension , minimum distance , and covering radius . If , it is a maximum distance separable (MDS) code. For an introduction to coding theory see [2, 16, 11, 19].

A *coset* of a code is a translation of the code. A coset of an code can be represented as

(1.1) |

where is a vector fixed for the given representation; see [2, 11, 16, 17, 19] and the references therein.

The weight distribution of code cosets is an important combinatorial property of a code. In particular, the distribution serves to estimate decoding results. There are many papers connected with distinct aspects of the weight distribution of cosets for codes over distinct fields and rings, see e.g.

[1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 13, 14, 20, 15, 21], [8, Sect. 6.3], [11, Sect. 7], [16, Sections 5.5, 6.6, 6.9], [17, Sect. 10] and the references therein.For a linear code of length , we call *integral weight spectrum* the overall numbers of weight vectors, , in all the cosets of a fixed weight.

*In this work,* for MDS codes, using and developing the results of [5], we obtain new convenient formulas of integral weight spectra of cosets of weight 1 and 2.
The obtained formulas for weight 1 and 2 cosets, seem to be simple and expressive.

This paper is organized as follows. Section 2 contains preliminaries. In Section 3, we consider the integral weight spectrum of the weight 1 cosets of MDS codes with minimum distance . In Section 4, we obtain the integral weight spectrum of the weight 2 cosets of MDS codes with minimum distance . In Section 5, we give the spectra for the weight 3 cosets of MDS codes with minimum distance and covering radius .

## 2 Preliminaries

### 2.1 Cosets of a linear code

We give a few known definitions and properties connected with cosets of linear codes, see e.g. [2, 11, 16, 17, 19] and the references therein.

We consider a coset of an code in the form (1.1). We have . One can take as any vector of . So, there are formally distinct representations of the form (1.1); all they give the same coset . If , we have . The distinct cosets of partition into sets of size .

We remind that the *Hamming weight* of the vector is the number of nonzero entries in .

###### Notation 2.1.

For an code and its coset of the form (1.1), the following notation is used:

the number of weight codewords of the code ; | |||

the weight of a coset | the smallest Hamming weight of any vector in the coset; | ||

a coset leader | a vector in the coset of the smallest Hamming weight; | ||

In cosets of weight , a vector of the minimal weight is not necessarily unique. Cosets of weight have a unique leader.

The code is the coset of weight zero. The leader of is the zero vector of .

###### Definition 2.2.

Let be an code and let be its coset of weight . Let be the overall number of weight vectors in all cosets of weight .
For a fixed , we call the set
*integral weight spectrum* of the code cosets of weight .

### 2.2 Some useful relations

For , the weight distribution of an MDS code has the following form, see e.g. [11, Th. 7.4.1], [16, Th. 11.3.6]:

(2.1) |

In , the volume of a sphere of radius is

(2.2) |

The following combinatorial identities are well known, see e.g. [18, Sect. 1, Eqs. (I),(IV), Problem 9(a)]:

(2.3) | |||

(2.4) | |||

(2.5) |

In [5, Eqs. (11)–(13)], for an MDS code correcting -fold errors, the following relations for denoted by are given:

(2.6) | |||

where | |||

(2.7) | |||

(2.8) | |||

## 3 The integral weight spectrum of the weight 1 cosets of MDS codes with minimum distance

In Sections 3–5, we represent the values

in distinct forms that can be convenient in distinct utilizations, e.g. for estimates of the decoder error probability, see

[5, 6] and the references therein.We use (with some transformations) the results of [5, Eqs. (11)–(13)] where, for an MDS code correcting -fold errors, the value gives the overall number of weight vectors in all cosets of weight . We cite [5, Eqs. (11)–(13)] by formulas (2.6)–(2.8), respectively.

In the rest of the paper we put that a sum is equal to zero if .

###### Lemma 3.1.

*[5, Eqs. (11)–(13)]*
Let . For an MDS code of minimum distance , the overall number of weight vectors in all cosets of weight is as follows:

(3.1) | |||

###### Proof.

###### Lemma 3.2.

The following holds:

(3.2) |

###### Lemma 3.3.

Let . The following holds:

For an code , we denote

(3.3) |

Also, we denote

(3.4) |

###### Theorem 3.4.

*(integral weight spectrum 1)*

Let . Let be an MDS code of minimum distance .

*(i)* For the code , the overall number of weight vectors in all weight cosets is as follows:

(3.5) | |||

(3.6) | |||

(3.7) | |||

(3.8) | |||

(3.9) |

*(ii)* Let the code be a MDS code of length and minimum distance . For , the overall number of weight vectors in all weight cosets is as follows

(3.10) | |||

*(iii)* Let the code be a MDS code of length and minimum distance . For , the overall number of weight vectors in all weight cosets is as follows

(3.11) |

###### Proof.

(i) By the definition of , see Notation 2.1, for the code of Lemma 3.1, we have

(3.12) |

We subtract (2.1) from (3.1) that gives

Here some simple transformations are missed out. Now, for the 2-nd sum , we use Lemma 3.2 and obtain (3.5).

For , we give a formula alternative to (3.1).

###### Corollary 3.5.

Let be as in (2.2). Let be an MDS code of minimum distance . Then for , the overall number of weight vectors in all cosets of weight is as follows:

(3.13) |

## 4 The integral weight spectrum of the weight 2 cosets of MDS codes with minimum distance

###### Lemma 4.1.

###### Proof.

For an code , we denote

(4.2) | |||

###### Lemma 4.2.

The following holds:

(4.3) |

###### Proof.

###### Theorem 4.3.

*(integral weight spectrum 2)*

*(i)* For the code , the overall number of weight vectors in all weight cosets is as follows:

(4.4) | |||

(4.5) | |||

(4.6) | |||

(4.7) | |||

(4.8) | |||

*(ii)* Let the code be a MDS code of length and minimum distance . For , the overall number of weight vectors in all weight cosets is as follows

(4.9) | |||

###### Proof.

(i) By the definition of , see Notation 2.1, for the code of Lemma 4.1, we have

(4.10) |

We subtract (3.1) from (4.1) that gives

Applying Lemma 3.2 to the 2-nd sum , after simple transformations we obtain

Due to (2.4) and (2.3), we have

Also, . Now we can obtain (4.4). Moreover, by (2.4), we have

that gives (4.5).

## 5 The integral weight spectrum of the weight 3 cosets of MDS codes with minimum distance and covering radius

###### Theorem 5.1.

*(integral weight spectrum 3)*

Let . Let be an MDS code of minimum distance and covering radius . Let , , , and be as in (2.2), (3.4), (4.1), and (4.2), respectively. Let
and be as in Theorems *3.4* and *4.3*, respectively.

*(i)* For the code , the overall number of weight vectors in all cosets of weight is as follows:

(5.1) | |||

(5.2) | |||

(5.3) | |||

*(ii)* Let the code be a MDS code of length , minimum distance , and covering radius . For , the overall number of weight vectors in all weight cosets is as follows

(5.4) | |||

(5.5) |

###### Proof.

(i) Due to covering radius , in there are not cosets of weight ; therefore for we have (5.1) where is the total number of weight vectors in .

## References

- [1] E.F. Assmus, Jr., H.F. Mattson, Jr., The weight-distribution of a coset of a linear code, IEEE Trans. Inform. Theory 24(4), 497 (1978) https://doi.org/10.1109/tit.1978.1055903
- [2] R.E. Blahut, Theory and Practice of Error Control Codes, Addison Wesley, Reading, 1984
- [3] A. Bonnecaze, I.M. Duursma, Translates of linear codes over , IEEE Trans. Inform. Theory 43(4), 1218–1230 (1997) https://doi.org/10.1109/18.605585
- [4] P. Charpin, T. Helleseth, V. Zinoviev, The coset distribution of triple-error-correcting binary primitive BCH codes. IEEE Trans. Inform. Theory, 52(4), 1727–1732 (2006)

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