A family of codes with locality containing optimal codes

Locally recoverable codes were introduced by Gopalan et al. in 2012, and in the same year Prakash et al. introduced the concept of codes with locality, which are a type of locally recoverable codes. In this work we introduce a new family of codes with locality, which are subcodes of a certain family of evaluation codes. We determine the dimension of these codes, and also bounds for the minimum distance. We present the true values of the minimum distance in special cases, and also show that elements of this family are "optimal codes", as defined by Prakash et al.

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1 Introduction

The class of locally recoverable codes was introduced in 2012 by Gopalan et al. (see [11]). The idea was to ensure reliable communication when using distributed storage systems. Thus the authors define a code as having locality if an entry at position of a codeword of length may be recovered from a set (which may vary with ) of at most other entries, for all . This would ensure the recovering of a codeword even in the presence of an erasure, due for example to a failure of some node in the network. In that same year Prakash et al. (see [13]) introduced the concept of codes with locality , also called -locally recoverable codes, which are codes of length such that for every position there is a subset containing and of size at most such that the -th entry of a codeword may be recovered from any subset of entries with positions in , so that we may recover any entry even with other erasures in the code.

In this paper we define a family consisting of subcodes of the so-called affine cartesian codes (see Definition 2.2) which are -locally recoverable codes. We call this family quasi affine cartesian codes. We determine their dimension (see Corollary 3.4 and Theorem 3.6) together with lower and upper bounds for the minimum distance (see Theorem 4.1). We list some cases where the codes are optimal (see Corollary 4.2) and we also determine the exact value of the minimum distance in some special cases of the code (see Theorem 4.1, Theorem 5.8 and Corollary 5.9).

In the next section we introduce the family of quasi affine cartesian codes, and prove that these codes are locally recoverable. In Section 3 we present several results on the dimension of these codes, after recalling some facts from Gröbner basis theory which we will need. In the following section we present lower and upper bounds for the minimum distance of quasi affine codes, and determine the exact values in some cases. We also prove that some of the codes we introduced are optimal codes. In Section 5 we treat a special case of quasi affine cartesian codes, for which we determine more values for the minimum distance. The paper ends with several numerical examples.

2 Quasi affine cartesian codes

Let be a finite field with elements.

Definition 2.1.

Let be positive integers, with and . We say that a (linear) code is -locally recoverable if for every there exists a subset , containing and of cardinality at most , such that the punctured code obtained by removing the entries which are not in has minimum distance at least .

The condition on the minimum distance in the above definition shows that one cannot have two distinct codewords in the punctured code which coincide in (at least) positions, so any positions in the set determine the remaining positions.

Let be a collection of non-empty subsets of , and let

 X:=K1×⋯×Kn:={(α1,…,αn)|αi∈Ki for all i}⊂Fnq.

Let for , so clearly , and let . It is not difficult to check that the ideal of polynomials in which vanish on is

 IX=⎛⎝∏α1∈K1(X1−α1),…,∏αn∈Kn(Xn−αn)⎞⎠

(see e.g. [12, Lemma 2.3] or [4, Lemma 3.11]). The evaluation morphism

 Ψ:Fq[X1,…,Xn]→Fmqf↦(f(α1),…,f(αm))

is an -linear map and . Actually, this is a surjective map because for each there exists a polynomial such that is equal to , if , or , if .

Let be a nonnegative integer. In what follows we will denote by the

-vector space formed by all polynomials of degree up to

, together with the zero polynomial.

Definition 2.2.

Let be a nonnegative integer. The affine cartesian code (of order ) defined over the sets is the image, by , of the polynomials in .

These codes appeared independently in [12] and [10] (in [10] in a generalized form). In the special case where we have the well-known generalized Reed-Muller code of order . In [12] the authors prove that we may ignore, in the cartesian product, sets with just one element and moreover may always assume that . The dimension and the minimum distance of these codes are known (see e.g. [12] or [10]).

In what follows we construct -locally recoverable codes which are subcodes of affine cartesian codes.

Definition 2.3.

Let and be integers with and , let and let be the set of polynomials such that , together with the zero polynomial. The -quasi affine cartesian code (of order ) defined over the sets is the image, by , of the set .

Theorem 2.4.

Let be subsets of such that for all , with , let be an integer such that and let be a nonnegative integer. For any , the -quasi affine cartesian code (of order ) defined over the sets is locally recoverable with locality where .

Proof.

Let , so . Let and let

 Iα={(α1,…αs−1,β,αs+1,…αn)∣β∈Ks},

a set which has elements. Assume that there exist such that we know the values , for , we will prove that then we can deduce the value of for any .

Write , where are polynomials in the variables , and let for . Denoting by the -th coordinate of , for , from the assumption we get that

 ck=f(βk)=r−1∑i=0biβik,% for k∈{1,…,r}.

This system of equations can be rewritten as a matrix equation

 ⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1β1β21⋯βr−111β2β22⋯βr−121β3β23⋯βr−131::⋱:1βrβ2r⋯βr−1r⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝b0b1b2:br−1⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝c1c2c3:cr⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,

which has a unique solution , since the square matrix is a Vandermonde matrix. This allow us to determine for any . ∎

3 On the dimension of quasi affine cartesian codes

In this section we determine the dimension of -quasi affine cartesian codes, and we will need some facts about Gröbner basis which we recall below.

Let be a monomial order in (the set of monomials of) , i.e.  is a total order, if then for all monomials , and 1 is the least monomial. The greatest monomial appearing in a polynomial is called the leading monomial of and is denoted by .

Definition 3.1.

Let be an ideal. A Gröbner basis (with respect to a monomial order ) for is a basis for such that the leading monomial of any polynomial in is a multiple of the leading monomial of some polynomial in . The footprint of (with respect to a monomial order ) is the set of monomials of which are not leading monomials of any polynomials in , and is denoted by .

B. Buchberger proved that, given a monomial order, any (nonzero) ideal admits a Gröbner basis (see [2] or [1, Sec. 1.7]). He also proved that a basis for as an -vector space is given by the classes of the monomials in (see e.g. [1, Prop. 2.1.6]).

Definition 3.2.

Let be a monomial order in and let be an ideal. Let be a (not necessarily Gröbner) basis for , we define as the set of monomials of which are not multiples of any of the leading monomials of .

Clearly we have and, moreover, if and only if is a Gröbner basis for .

In what follows we will use the graded-lexicographic order in , with .

For let , so that and . Since any two of the leading monomials of are coprime we get that is a Gröbner basis for (see [9, Prop. 4, page 104]) so

 Δ(IX)=Δ(Xd11,…,Xdnn)={Xa11⋯Xann∣0≤ai

Let , it is known (see e.g. [4, Prop. 3.12]) that . This implies that if then , while if then

 dim(CX(d))=(n+dd)−n∑i=1(n+d−did−di)+⋯+(−1)j∑1≤i1<⋯

where we set if .

To determine the dimension of we make a reasoning similar to the one used to prove the above formulas.

Let , then .

Proof.

Given let be its remainder in the division by , then . From the division algorithm we know that any monomial which appear in is not a multiple of for all , and also that and . Thus and moreover, is a linear combination of monomials in . This shows that . Let

 ¯¯¯¯Ψ:Fq[X1,…,Xn]/IX→Fmq

be defined as , we know that is an isomorphism and clearly , where is the -vector space generated by the monomials in . Since we know from Buchberger’s result that the classes in of the monomials in are linearly independent over , thus we get . ∎

Let .

Corollary 3.4.

If then , and

 dim(D(δ,s)X(~d))=(ds−δ+1)n∏i=1i≠sdi.

Also .

Proof.

From the above proof we get that if then and for all . Thus if we have which implies . We also have that

 dim(D(δ,s)X(~d))=|{Xa11⋯Xann∣0≤ai

Observe that in there is only one monomial of degree , so that . ∎

Now, for (when we have ), we present a formula for the dimension of in terms of the dimension of certain affine cartesian codes, and for that we introduce some notation.

Definition 3.5.

For we denote by the product

 Xs=K1×⋯×Ks−1×Ks+1×⋯×Kn.

Observe that we may define the affine cartesian code as in Definition 2.2, except that now is defined over the sets .

Theorem 3.6.

Let and let be an integer such that . If then , and if then

 dimFqD(δ,s)X(d)=dimFqCX(d)−δ−2∑i=0dimFqCXs(d−r−i), (1)

where if .

Proof.

If then from Definitions 2.2 and 2.3 we get that , so we assume now that . Define the following sets:

 Ωd = {(a1,…,an)∈Nn∣0≤ai

From previous considerations we get that and . For any we have that either or for some in the range (because ). If then we have

 a1+⋯+as−1+as+1+⋯+an≤d−r−i,

and for we define

 Ω(0)s,d−r−i={(a1,…,an)∈Ωd−r−i∣as=0},

so that if is such that . Thus we have

 |Ωd|=|Ω(δ,s)d|+δ−2∑i=0|Ω(0)s,d−r−i|

and since for all the above equation implies equation (1) in the statement. ∎

4 Minimum distance and optimal codes

In this section we relate the minimum distance of quasi affine cartesian codes to the minimum distance of affine cartesian codes. In what follows we denote by the minimum distance of a code .

Let be an integer in the range , and let and be uniquely defined by writing , with (if then take and , if then we understand that ). We recall that

 W(1)(CX(d))=(dk+1−ℓ)n∏i=k+2di (2)

(see e.g. [12, Theorem 3.8]).

Theorem 4.1.

Let where and . We have

 W(1)(CX(d))≤W(1)(D(δ,s)X(d))≤m−dimFqD(δ,s)X(d)−⎛⎝⎡⎢ ⎢ ⎢⎢dimFqD(δ,s)X(d)r⎤⎥ ⎥ ⎥⎥−1⎞⎠(δ−1)+1. (3)

where , and for , is the smallest integer such that . If

1. and , or

2. and

then we get .

Proof.

Since , we have . From Theorem 2.4 we know that is locally recoverable with locality , so we may apply [13, Theorem 2] and we get the second inequality of (3).

Assume that and . We consider two cases, and , let’s suppose first that . Consider an element , and consider distinct elements . Define the polynomial

 f=k∏i=1∏α∈Kiα≠αi(Xi−α)⋅ℓ∏i=1(Xk+1−βi). (4)

Observe that so that . Denoting by the weight of a codeword we have , and we’re done. Assume now that , from we must have . Clearly so replacing by in (4) we still have and .

Finally suppose that is satisfied, i.e.  and , and to avoid overlapping with the previous case we also assume that either or . Now we take

 f=k+1∏i=1i≠s∏α∈Kiα≠αi(Xi−α)⋅ds−(dk+1−ℓ)∏i=1(Xs−βi),

where are distinct elements of , and again we have , and . ∎

Following [13], we say that the code is optimal if its minimum distance attains the upper bound presented in the above theorem.

Corollary 4.2.

The codes and are optimal, and have minimum distance equal to, respectively, and .

Proof.

We have so from (2) we get . On the other hand, from Corollary 3.4 and the fact that we get that the upper bound for in the above theorem is

 m−dimFqD(δ,s)X(d)−⎛⎝⎡⎢ ⎢ ⎢⎢dimFqD(δ,s)X(d)r⎤⎥ ⎥ ⎥⎥−1⎞⎠(δ−1)+1=n∏i=1di−(ds−δ+1)n∏i=1i≠sdi−⎛⎜ ⎜⎝n∏i=1i≠sdi−1⎞⎟ ⎟⎠(δ−1)+1=δ (5)

so . In the same way one proves that . ∎

One may check that if and then either condition or condition of the above Proposition is satisfied, so we get . In the following section, among other results, we present some values for when we have .

5 Further results on the minimum distance in a special case

In this section we assume that are fields such that .

We write for the affine group of , i.e. the transformations of of the type , where and .

Definition 5.1.

The affine group associated to is

 Aff\/(X)={φ:X→X∣φ=ψ|X with ψ∈Aff\/(Fnq) and ψ(X)=X}.

Let be the canonical basis of , since we get that for each there exists only one such that .

Lemma 5.2.

Let be given by , where

 A=⎛⎜ ⎜⎝a11⋯a1n:⋱:an1⋯ann⎞⎟ ⎟⎠andβ=⎛⎜⎝b1:bn⎞⎟⎠,

and let . Then if and only if the following conditions are satisfied:

1. for all , , and if then ;

2. for all such that the square submatrix formed by entries with