A family of codes with locality containing optimal codes

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 $r$ if an entry at position $i$ of a codeword of length $n$ may be recovered from a set (which may vary with $i$ ) of at most $r$ other entries, for all $i = 1, \dots, n$ . 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 $(r, δ)$ , also called $(r, δ)$ -locally recoverable codes, which are codes of length $n$ such that for every position $i \in {1, \dots, n}$ there is a subset $S_{i} \subset {1, \dots, n}$ containing $i$ and of size at most $r + δ - 1$ such that the $i$ -th entry of a codeword may be recovered from any subset of $r$ entries with positions in $S_{i} ∖ {i}$ , so that we may recover any entry even with $δ - 2$ 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 $(r, δ)$ -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 $F_{q}$ be a finite field with $q$ elements.

Definition 2.1.

Let $m, r, δ$ be positive integers, with $δ \geq 2$ and $r + δ - 1 \leq m$ . We say that a (linear) code $C \subset F_{q}^{m}$ is $(r, δ)$ -locally recoverable if for every $i \in {1, \dots, m}$ there exists a subset $S_{i} \subset {1, \dots, n}$ , containing $i$ and of cardinality at most $r + δ - 1$ , such that the punctured code obtained by removing the entries which are not in $S_{i}$ has minimum distance at least $δ$ .

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) $r$ positions, so any $r$ positions in the set $S_{i}$ determine the remaining $δ - 1$ positions.

Let $K_{1}, \dots, K_{n}$ be a collection of non-empty subsets of $F_{q}$ , and let

X := K_{1} \times \dots \times K_{n} := {(α_{1}, \dots, α_{n}) | α_{i} \in K_{i} for all i} \subset F_{q}^{n} .

Let $d_{i} := | K_{i} |$ for $i = 1, \dots, n$ , so clearly $| X | = \prod_{i = 1}^{n} d_{i} =: m$ , and let $X = {α_{1}, \dots, α_{m}}$ . It is not difficult to check that the ideal of polynomials in $F_{q} [X_{1}, \dots, X_{n}]$ which vanish on $X$ is

I_{X} = ⎛ ⎝ \prod α_{1} \in K_{1} (X_{1} - α_{1}), \dots, \prod α_{n} \in K_{n} (X_{n} - α_{n}) ⎞ ⎠

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

\begin{matrix} Ψ : F_{q} [X_{1}, \dots, X_{n}] & \to & F_{q}^{m} f & \mapsto & (f (α_{1}), \dots, f (α_{m})) \end{matrix}

is an $F_{q}$ -linear map and $ker Ψ = I_{X}$ . Actually, this is a surjective map because for each $i \in {1, \dots, m}$ there exists a polynomial $f_{i}$ such that $f_{i} (α_{j})$ is equal to $1$ , if $j = i$ , or $0$ , if $j \neq i$ .

Let $d$ be a nonnegative integer. In what follows we will denote by $F_{q} [X_{1}, \dots, X_{n}]_{\leq d}$ the $F_{q}$

-vector space formed by all polynomials of degree up to

d

, together with the zero polynomial.

Definition 2.2.

Let $d$ be a nonnegative integer. The affine cartesian code (of order $d$ ) $C_{X} (d)$ defined over the sets $K_{1}, \dots, K_{n}$ is the image, by $Ψ$ , of the polynomials in $F_{q} [X_{1}, \dots, X_{n}]_{\leq d}$ .

These codes appeared independently in [12] and [10] (in [10] in a generalized form). In the special case where $K_{1} = \dots = K_{n} = F_{q}$ we have the well-known generalized Reed-Muller code of order $d$ . In [12] the authors prove that we may ignore, in the cartesian product, sets with just one element and moreover may always assume that $2 \leq d_{1} \leq \dots \leq d_{n}$ . The dimension and the minimum distance of these codes are known (see e.g. [12] or [10]).

In what follows we construct $(r, δ)$ -locally recoverable codes which are subcodes of affine cartesian codes.

Definition 2.3.

Let $d$ and $δ$ be integers with $d \geq 0$ and $δ \geq 2$ , let $s \in {1, \dots, n}$ and let $P_{d}^{(δ, s)}$ be the set of polynomials $f \in F_{q} [X_{1}, \dots, X_{n}]_{\leq d}$ such that ${deg}_{X_{s}} f < d_{s} - δ + 1$ , together with the zero polynomial. The $(δ, s)$ -quasi affine cartesian code (of order $d$ ) $D_{X}^{(δ, s)} (d)$ defined over the sets $K_{1}, \dots, K_{n}$ is the image, by $Ψ$ , of the set $P_{d}^{(δ, s)}$ .

Theorem 2.4.

Let $K_{1}, \dots, K_{n}$ be subsets of $F_{q}$ such that $| K_{i} | = d_{i} \geq 2$ for all $i = 1, \dots, n$ , with $n \geq 2$ , let $δ \geq 2$ be an integer such that $d_{s} - δ + 1 \geq 1$ and let $d$ be a nonnegative integer. For any $s \in {1, \dots, n}$ , the $(δ, s)$ -quasi affine cartesian code (of order $d$ ) $D_{X}^{(δ, s)} (d)$ defined over the sets $K_{1}, \dots, K_{n}$ is locally recoverable with locality $(r, δ)$ where $r = d_{s} - δ + 1$ .

Proof.

Let $f \in P_{d}^{(δ, s)}$ , so $(f (α_{1}), \dots, f (α_{m})) \in D_{X}^{(δ, s)} (d)$ . Let $α = (α_{1}, \dots, α_{n}) \in X$ and let

I_{α} = {(α_{1}, \dots α_{s - 1}, β, α_{s + 1}, \dots α_{n}) ∣ β \in K_{s}},

a set which has $d_{s} = r + δ - 1$ elements. Assume that there exist $β_{1}, \dots, β_{r} \in I_{α}$ such that we know the values $f (β_{k}) =: c_{k}$ , for $k \in {1, \dots, r}$ , we will prove that then we can deduce the value of $f (β)$ for any $β \in I_{α}$ .

Write $f = \sum_{i = 0}^{r - 1} g_{i} X_{s}^{i}$ , where $g_{1}, \dots, g_{r - 1}$ are polynomials in the variables $X_{1}, \dots, X_{s - 1}, X_{s + 1}, \dots, X_{n}$ , and let $b_{i} := g_{i} (α_{1}, \dots α_{s - 1}, α_{s + 1}, \dots α_{n})$ for $i = 0, \dots, r - 1$ . Denoting by $β_{k}$ the $s$ -th coordinate of $β_{k}$ , for $k = 1, \dots, r$ , from the assumption we get that

c_{k} = f (β_{k}) = r - 1 \sum i = 0 b_{i} β_{k}^{i}, % for k \in {1, \dots, r} .

This system of equations can be rewritten as a matrix equation

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} 1 & β_{1} & β_{1}^{2} & \dots & β_{1}^{r - 1} 1 & β_{2} & β_{2}^{2} & \dots & β_{2}^{r - 1} 1 & β_{3} & β_{3}^{2} & \dots & β_{3}^{r - 1} 1 & : & : & ⋱ & : 1 & β_{r} & β_{r}^{2} & \dots & β_{r}^{r - 1} \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} b_{0} b_{1} b_{2} : b_{r - 1} \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ \begin{matrix} c_{1} c_{2} c_{3} : c_{r} \end{matrix} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠,

which has a unique solution $(b_{0}, b_{1}, \dots, b_{r - 1})$ , since the square $r \times r$ matrix is a Vandermonde matrix. This allow us to determine $f (β)$ for any $β \in I_{α}$ . ∎

3 On the dimension of quasi affine cartesian codes

In this section we determine the dimension of $(δ, s)$ -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) $F_{q} [X_{1}, \dots, X_{n}]$ , i.e. $≺$ is a total order, if $M_{1} ≺ M_{2}$ then $M M_{1} ≺ M M_{2}$ for all monomials $M, M_{1}, M_{2}$ , and 1 is the least monomial. The greatest monomial appearing in a polynomial $f$ is called the leading monomial of $f$ and is denoted by $L M (f)$ .

Definition 3.1.

Let $J \subset F_{q} [X_{1}, \dots, X_{n}]$ be an ideal. A Gröbner basis (with respect to a monomial order $≺$ ) for $J$ is a basis $G$ for $J$ such that the leading monomial of any polynomial in $J$ is a multiple of the leading monomial of some polynomial in $G$ . The footprint of $J$ (with respect to a monomial order $≺$ ) is the set of monomials of $F_{q} [X_{1}, \dots, X_{n}]$ which are not leading monomials of any polynomials in $J$ , and is denoted by $Δ (J)$ .

B. Buchberger proved that, given a monomial order, any (nonzero) ideal $J \subset F_{q} [X_{1}, \dots, X_{n}]$ admits a Gröbner basis (see [2] or [1, Sec. 1.7]). He also proved that a basis for $F_{q} [X_{1}, \dots, X_{n}] / J$ as an $F_{q}$ -vector space is given by the classes of the monomials in $Δ (J)$ (see e.g. [1, Prop. 2.1.6]).

Definition 3.2.

Let $≺$ be a monomial order in $F_{q} [X_{1}, \dots, X_{n}]$ and let $J \subset F_{q} [X_{1}, \dots, X_{n}]$ be an ideal. Let ${g_{1}, \dots, g_{r}}$ be a (not necessarily Gröbner) basis for $J$ , we define $Δ (L M (g_{1}), \dots, L M (g_{r}))$ as the set of monomials of $F_{q} [X_{1}, \dots, X_{n}]$ which are not multiples of any of the leading monomials of $g_{1}, \dots, g_{r}$ .

Clearly we have $Δ (J) \subset Δ (L M (g_{1}), \dots, L M (g_{r}))$ and, moreover, $Δ (J) = Δ (L M (g_{1}), \dots, L M (g_{r}))$ if and only if ${g_{1}, \dots, g_{r}}$ is a Gröbner basis for $J$ .

In what follows we will use the graded-lexicographic order in $F_{q} [X_{1}, \dots, X_{n}]$ , with $X_{n} ≺ \dots ≺ X_{1}$ .

For $i = 1, \dots, n$ let $f_{i} = \prod_{α \in K_{i}} (X_{i} - α)$ , so that $deg f_{i} = d_{i}$ and $I_{X} = ⟨ f_{1}, \dots, f_{n} ⟩$ . Since any two of the leading monomials of $f_{1}, \dots, f_{n}$ are coprime we get that ${f_{1}, \dots, f_{n}}$ is a Gröbner basis for $I_{X}$ (see [9, Prop. 4, page 104]) so

Δ (I_{X}) = Δ (X_{1}^{d_{1}}, \dots, X_{n}^{d_{n}}) = {X_{1}^{a_{1}} \dots X_{n}^{a_{n}} ∣ 0 \leq a_{i} < d_{i}, \forall i = 1, \dots, n} .

Let $Δ (I_{X})_{\leq d} = {M \in Δ (I_{X}) ∣ deg (M) \leq d}$ , it is known (see e.g. [4, Prop. 3.12]) that $dim (C_{X} (d)) = | Δ (I_{X})_{\leq d} |$ . This implies that if $d \geq \sum_{i = 1}^{n} (d_{i} - 1)$ then $dim (C_{X} (d)) = | Δ (I_{X})_{\leq d} | = | Δ (I) | = \prod_{i = 1}^{n} d_{i}$ , while if $0 \leq d < \prod_{i = 1}^{n} d_{i}$ then

\begin{matrix} dim (C_{X} (d)) = (\frac{n + d}{d}) - n \sum i = 1 (\frac{n + d - d_{i}}{d - d_{i}}) + \dots + (- 1)^{j} \sum 1 \leq i_{1} < \dots < i_{j} \leq n (\frac{n + d - d_{i_{1}} - \dots - d_{i_{j}}}{d - d_{i_{1}} - \dots - d_{i_{j}}}) + \dots + (- 1)^{n} (\frac{n + d - d_{1} - \dots - d_{n}}{d - d_{1} - \dots - d_{n}}) \end{matrix}

where we set $(\frac{a}{b}) = 0$ if $b < 0$ .

To determine the dimension of $D_{X}^{(δ, s)} (d)$ we make a reasoning similar to the one used to prove the above formulas.

Proposition 3.3.

Let $Δ (I_{X})_{\leq d}^{(δ, s)} = {M \in Δ (I_{X})_{\leq d} ∣ {deg}_{X_{s}} M < d_{s} - δ + 1}$ , then $dim (D_{X}^{(δ, s)} (d)) = | Δ (I_{X})_{\leq d}^{(δ, s)} |$ .

Proof.

Given $f \in P_{d}^{(δ, s)}$ let $g \in F_{q} [X_{1}, \dots, X_{n}]$ be its remainder in the division by ${f_{1}, \dots, f_{n}}$ , then $Ψ (f) = Ψ (g)$ . From the division algorithm we know that any monomial which appear in $g$ is not a multiple of $L M (f_{i}) = X_{i}^{d_{i}}$ for all $i = 1, \dots, n$ , and also that $deg g \leq deg f$ and ${deg}_{X_{s}} g < d_{s} - δ + 1$ . Thus $g \in P_{d}^{(δ, s)}$ and moreover, $g$ is a linear combination of monomials in $Δ (I_{X})_{\leq d}^{(δ, s)}$ . This shows that $dim (D_{X}^{(δ, s)} (d)) \leq | Δ (I_{X})_{\leq d}^{(δ, s)} |$ . Let

¯ ¯¯ ¯ Ψ : F_{q} [X_{1}, \dots, X_{n}] / I_{X} \to F_{q}^{m}

be defined as $¯ ¯¯ ¯ Ψ (f + I_{X}) = Ψ (f)$ , we know that $¯ ¯¯ ¯ Ψ$ is an isomorphism and clearly $D_{X}^{(δ, s)} (d) = {¯ ¯¯ ¯ Ψ (h + I_{X}) ∣ h \in ⟨ Δ (I_{X})_{\leq d}^{(δ, s)} ⟩}$ , where $⟨ Δ (I_{X})_{\leq d}^{(δ, s)} ⟩$ is the $F_{q}$ -vector space generated by the monomials in $Δ (I_{X})_{\leq d}^{(δ, s)}$ . Since $Δ (I_{X})_{\leq d}^{(δ, s)} \subset Δ (I_{X})$ we know from Buchberger’s result that the classes in $F_{q} [X_{1}, \dots, X_{n}] / I_{X}$ of the monomials in $Δ (I_{X})_{\leq d}^{(δ, s)}$ are linearly independent over $F_{q}$ , thus we get $dim (D_{X}^{(δ, s)} (d)) = | Δ (I_{X})_{\leq d}^{(δ, s)} |$ . ∎

Let $~ d := n \sum \begin{matrix} i = 1 i \neq s \end{matrix} (d_{i} - 1) + d_{s} - δ$ .

Corollary 3.4.

If $d \geq ~ d$ then $D_{X}^{(δ, s)} (d) = D_{X}^{(δ, s)} (~ d)$ , and

dim (D_{X}^{(δ, s)} (~ d)) = (d_{s} - δ + 1) n \prod \begin{matrix} i = 1 i \neq s \end{matrix} d_{i} .

Also $dim (D_{X}^{(δ, s)} (~ d - 1)) = dim (D_{X}^{(δ, s)} (~ d)) - 1$ .

Proof.

From the above proof we get that if $M \in Δ (I_{X})_{\leq d}^{(δ, s)}$ then ${deg}_{X_{s}} (M) < d_{s} - δ + 1$ and ${deg}_{X_{i}} (M) < d_{i}$ for all $i \in {1, \dots, n} ∖ {s}$ . Thus if $d \geq ~ d$ we have $Δ (I_{X})_{\leq d}^{(δ, s)} = Δ (I_{X})_{\leq ~ d}^{(δ, s)}$ which implies $D_{X}^{(δ, s)} (d) = D_{X}^{(δ, s)} (~ d)$ . We also have that

\begin{matrix} dim (D_{X}^{(δ, s)} (~ d)) = | {X_{1}^{a_{1}} \dots X_{n}^{a_{n}} ∣ 0 \leq a_{i} < d_{i}, i = 1, \dots, n, i \neq s, and 0 \leq a_{s} < d_{s} - δ + 1} | = (d_{s} - δ + 1) n \prod \begin{matrix} i = 1 i \neq s \end{matrix} d_{i} \end{matrix}

Observe that in $Δ (I_{X})_{\leq ~ d}^{(δ, s)}$ there is only one monomial of degree $~ d$ , so that $dim (D_{X}^{(δ, s)} (~ d - 1)) = dim (D_{X}^{(δ, s)} (~ d)) - 1$ . ∎

Now, for $1 \leq d < ~ d$ (when $d = 0$ we have $D^{(δ, s)} (0) ≃ F_{q}$ ), we present a formula for the dimension of $D^{(δ, s)} (d)$ in terms of the dimension of certain affine cartesian codes, and for that we introduce some notation.

Definition 3.5.

For $s \in {1, \dots, n}$ we denote by $X_{s}$ the product

X_{s} = K_{1} \times \dots \times K_{s - 1} \times K_{s + 1} \times \dots \times K_{n} .

Observe that we may define the affine cartesian code $C_{X_{s}} (d)$ as in Definition 2.2, except that now $C_{X_{s}} (d)$ is defined over the sets $K_{1}, \dots, K_{s - 1},$ $K_{s + 1}, \dots, K_{n}$ .

Theorem 3.6.

Let $s \in {1, \dots, n}$ and let $d$ be an integer such that $1 \leq d < ~ d$ . If $1 \leq d < r = d_{s} - δ + 1$ then ${dim}_{F_{q}} D_{X}^{(δ, s)} (d) = {dim}_{F_{q}} C_{X} (d)$ , and if $r \leq d \leq ~ d$ then

{dim}_{F_{q}} D_{X}^{(δ, s)} (d) = {dim}_{F_{q}} C_{X} (d) - δ - 2 \sum i = 0 {dim}_{F_{q}} C_{X_{s}} (d - r - i),

(1)

where ${dim}_{F_{q}} C_{X_{s}} (d - r - i) = 0$ if $d - r - i < 0$ .

Proof.

If $1 \leq d < r = d_{s} - δ + 1$ then from Definitions 2.2 and 2.3 we get that ${dim}_{F_{q}} D_{X}^{(δ, s)} (d) = {dim}_{F_{q}} C_{X} (d)$ , so we assume now that $r \leq d \leq ~ d$ . Define the following sets:

	$Ω_{d}$	$=$	${(a_{1}, \dots, a_{n}) \in N^{n} ∣ 0 \leq a_{i} < d_{i}, for 1 \leq i \leq n, a_{1} + \dots + a_{n} \leq d};$
	$Ω_{d}^{(δ, s)}$	$=$	${(a_{1}, \dots, a_{n}) \in Ω_{d} ∣ a_{s} \leq d_{s} - δ} .$

From previous considerations we get that ${dim}_{F_{q}} C_{X} (d) = | Ω_{d} |$ and ${dim}_{F_{q}} D_{X}^{(δ, s)} (d) = | Ω_{d}^{(δ, s)} |$ . For any $(a_{1}, \dots, a_{n}) \in Ω_{d}$ we have that either $a_{s} \leq d_{s} - δ$ or $a_{s} = d_{s} - δ + 1 + i$ for some $i$ in the range $0 \leq i \leq δ - 2$ (because $a_{s} \leq d_{s} - 1$ ). If $a_{s} = d_{s} - δ + 1 + i = r + i$ then we have

a_{1} + \dots + a_{s - 1} + a_{s + 1} + \dots + a_{n} \leq d - r - i,

and for $0 \leq i \leq δ - 2$ we define

Ω_{s, d - r - i}^{(0)} = {(a_{1}, \dots, a_{n}) \in Ω_{d - r - i} ∣ a_{s} = 0},

so that $Ω_{s, d - r - i}^{(0)} = \emptyset$ if $i$ is such that $d - r - i < 0$ . Thus we have

| Ω_{d} | = | Ω_{d}^{(δ, s)} | + δ - 2 \sum i = 0 | Ω_{s, d - r - i}^{(0)} |

and since ${dim}_{F_{q}} C_{X_{s}} (d - r - i) = | Ω_{s, d - r - i}^{(0)} |$ for all $i \in {0, \dots, δ - 2}$ 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 $W^{(1)} (C)$ the minimum distance of a code $C$ .

Let $d$ be an integer in the range $1 \leq d < \sum_{i = 1}^{n} (d_{i} - 1)$ , and let $k$ and $ℓ$ be uniquely defined by writing $d = \sum_{i = 1}^{k} (d_{i} - 1) + ℓ$ , with $0 < ℓ \leq d_{k + 1} - 1$ (if $d < d_{1} - 1$ then take $k = 0$ and $ℓ = d$ , if $k + 1 = n$ then we understand that $\prod_{i = k + 2}^{n} d_{i} = 1$ ). We recall that

W^{(1)} (C_{X} (d)) = (d_{k + 1} - ℓ) n \prod i = k + 2 d_{i}

(2)

(see e.g. [12, Theorem 3.8]).

Theorem 4.1.

Let $d = k \sum i = 1 (d_{i} - 1) + ℓ$ where $0 \leq k < n$ and $0 < ℓ \leq d_{k + 1} - 1$ . We have

W^{(1)} (C_{X} (d)) \leq W^{(1)} (D_{X}^{(δ, s)} (d)) \leq m - {dim}_{F_{q}} D_{X}^{(δ, s)} (d) - ⎛ ⎝ ⎡ ⎢ ⎢ ⎢ ⎢ \frac{{dim}_{F_{q}} D_{X}^{(δ, s)} (d)}{r} ⎤ ⎥ ⎥ ⎥ ⎥ - 1 ⎞ ⎠ (δ - 1) + 1.

(3)

where $m = \prod_{i = 1}^{n} d_{i}$ , $r = d_{s} - δ + 1$ and for $x \in R$ , $⌈ x ⌉$ is the smallest integer such that $x \leq ⌈ x ⌉$ . If

$k + 2 \leq n$ and $d_{k + 2} \leq d_{s}$ , or
$d_{s} \leq d_{k + 1}$ and $0 \leq d_{s} - (d_{k + 1} - ℓ) < r$

then we get $W^{(1)} (D_{X}^{(δ, s)} (d)) = W^{(1)} (C_{X} (d))$ .

Proof.

Since $D_{X}^{(δ, s)} (d) \subset C_{X} (d)$ , we have $W^{(1)} (C_{X} (d)) \leq W^{(1)} (D_{X}^{(δ, s)} (d))$ . From Theorem 2.4 we know that $D_{X}^{(δ, s)} (d)$ is locally recoverable with locality $(r, δ)$ , so we may apply [13, Theorem 2] and we get the second inequality of (3).

Assume that $k + 2 \leq n$ and $d_{k + 2} \leq d_{s}$ . We consider two cases, $s \geq k + 2$ and $s < k + 2$ , let’s suppose first that $s \geq k + 2$ . Consider an element $α = (α_{1}, \dots, α_{n}) \in X$ , and consider distinct elements $β_{1}, \dots, β_{ℓ} \in K_{k + 1}$ . Define the polynomial

f = k \prod i = 1 \prod \begin{matrix} α \in K_{i} α \neq α_{i} \end{matrix} (X_{i} - α) \cdot ℓ \prod i = 1 (X_{k + 1} - β_{i}) .

(4)

Observe that $f \in P_{d}^{(δ, s)}$ so that $Ψ (f) \in D_{X}^{(δ, s)} (d)$ . Denoting by $w (v)$ the weight of a codeword $v$ we have $w (Ψ (f)) = W^{(1)} (C_{X} (d))$ , and we’re done. Assume now that $s < k + 2$ , from $d_{k + 2} \leq d_{s}$ we must have $K_{s} = K_{k + 1} = K_{k + 2}$ . Clearly $s \in {1, \dots, k + 1}$ so replacing $K_{s}$ by $K_{k + 2}$ in (4) we still have $f \in P_{d}^{(δ, s)}$ and $w (Ψ (f)) = W^{(1)} (C_{X} (d))$ .

Finally suppose that $(i i)$ is satisfied, i.e. $s \leq k + 1$ and $0 \leq d_{s} - (d_{k + 1} - ℓ) < r$ , and to avoid overlapping with the previous case we also assume that either $d_{s} < d_{k + 2}$ or $n = k + 1$ . Now we take

f = k + 1 \prod \begin{matrix} i = 1 i \neq s \end{matrix} \prod \begin{matrix} α \in K_{i} α \neq α_{i} \end{matrix} (X_{i} - α) \cdot d_{s} - (d_{k + 1} - ℓ) \prod i = 1 (X_{s} - β_{i}),

where $β_{1}, \dots, β_{d_{s} - (d_{k + 1} - ℓ)}$ are distinct elements of $K_{s}$ , and again we have $f \in P_{d}^{(δ, s)}$ , $Ψ (f) \in D_{X}^{(δ, s)} (d)$ and $w (Ψ (f)) = W^{(1)} (C_{X} (d))$ . ∎

Following [13], we say that the code $D_{X}^{(δ, s)} (d)$ is optimal if its minimum distance attains the upper bound presented in the above theorem.

Corollary 4.2.

The codes $D_{X}^{(δ, s)} (~ d)$ and $D_{X}^{(δ, s)} (~ d - 1)$ are optimal, and have minimum distance equal to, respectively, $δ$ and $δ + 1$ .

Proof.

We have $~ d = n \sum \begin{matrix} i = 1 i \neq s \end{matrix} (d_{i} - 1) + d_{s} - δ = n - 1 \sum i = 1 (d_{i} - 1) + d_{n} - δ$ so from (2) we get $W^{(1)} (C_{X} (~ d)) = d_{n} - (d_{n} - δ) = δ$ . On the other hand, from Corollary 3.4 and the fact that $r = d_{s} - δ + 1$ we get that the upper bound for $W^{(1)} (D_{X}^{(δ, s)} (~ d))$ in the above theorem is

m - {dim}_{F_{q}} D_{X}^{(δ, s)} (d) - ⎛ ⎝ ⎡ ⎢ ⎢ ⎢ ⎢ \frac{{dim}_{F_{q}} D_{X}^{(δ, s)} (d)}{r} ⎤ ⎥ ⎥ ⎥ ⎥ - 1 ⎞ ⎠ (δ - 1) + 1 = n \prod i = 1 d_{i} - (d_{s} - δ + 1) n \prod \begin{matrix} i = 1 i \neq s \end{matrix} d_{i} - ⎛ ⎜ ⎜ ⎝ n \prod \begin{matrix} i = 1 i \neq s \end{matrix} d_{i} - 1 ⎞ ⎟ ⎟ ⎠ (δ - 1) + 1 = δ

(5)

so $W^{(1)} (D_{X}^{(δ, s)} (~ d)) = δ$ . In the same way one proves that $W^{(1)} (D_{X}^{(δ, s)} (~ d - 1)) = δ + 1$ . ∎

One may check that if $d_{s} = d_{n}$ and $d \leq ~ d$ then either condition $(i)$ or condition $(i i)$ of the above Proposition is satisfied, so we get $W^{(1)} (D_{X}^{(δ, s)} (d)) = W^{(1)} (C_{X} (d))$ . In the following section, among other results, we present some values for $W^{(1)} (D_{X}^{(δ, s)} (d))$ when we have $W^{(1)} (D_{X}^{(δ, s)} (d)) > W^{(1)} (C_{X} (d))$ .

5 Further results on the minimum distance in a special case

In this section we assume that $K_{1}, \dots, K_{n}$ are fields such that $K_{1} \subset K_{2} \subset \dots \subset K_{n} \subset F_{q}$ .

We write $Aff\/(Fnq)$ for the affine group of $F_{q}^{n}$ , i.e. the transformations of $F_{q}^{n}$ of the type $α ⟼ A α + β$ , where $A \in G L (n, F_{q})$ and $β \in F_{q}^{n}$ .

Definition 5.1.

The affine group associated to $X$ is

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

Let ${e_{1}, \dots, e_{n}} \subset F_{q}^{n}$ be the canonical basis of $F_{q}^{n}$ , since $e_{1}, \dots, e_{n} \in X$ we get that for each $φ∈Aff\/(X)$ there exists only one $ψ∈Aff\/(Fnq)$ such that $φ = ψ_{|_{X}}$ .

Lemma 5.2.

Let $ψ∈Aff\/(Fnq)$ be given by $α ⟼ A α + β$ , where

A = ⎛ ⎜ ⎜ ⎝ \begin{matrix} a_{11} & \dots & a_{1 n} : & ⋱ & : a_{n 1} & \dots & a_{n n} \end{matrix} ⎞ ⎟ ⎟ ⎠ and β = ⎛ ⎜ ⎝ \begin{matrix} b_{1} : b_{n} \end{matrix} ⎞ ⎟ ⎠,

and let $φ = ψ_{|_{X}}$ . Then $φ∈Aff\/(X)$ if and only if the following conditions are satisfied:

for all $i, j \in {1, \dots, n}$ , $a_{i j} \in K_{i}$ , $b_{j} \in K_{j}$ and if $K_{i} ⫋ K_{j}$ then $a_{i j} = 0$ ;
for all $i \leq j \in {1, \dots, n}$ such that $K_{i - 1} ⫋ K_{i} = K_{j} ⫋ K_{j + 1}$ the square submatrix formed by entries $a_{u w}$ with $i \leq u, w \leq$

A family of codes with locality containing optimal codes

Authors

Cyclic Codes with Locality and Availability

An axiomatization of Λ-quantiles

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A Study of the Separating Property in Reed-Solomon Codes by Bounding the Minimum Distance

Dynamic Local Searchable Symmetric Encryption

New Constructions of Optimal Cyclic (r,δ) Locally Repairable Codes from Their Zeros

Optimal quaternary (r,δ)-Locally Recoverable Codes: Their Structures and Complete Classification

1 Introduction

2 Quasi affine cartesian codes

Definition 2.1.

Definition 2.2.

Definition 2.3.

Theorem 2.4.

Proof.

3 On the dimension of quasi affine cartesian codes

Definition 3.1.

Definition 3.2.

Proposition 3.3.

Proof.

Corollary 3.4.

Proof.

Definition 3.5.

Theorem 3.6.

Proof.

4 Minimum distance and optimal codes

Theorem 4.1.

Proof.

Corollary 4.2.

Proof.

5 Further results on the minimum distance in a special case

Definition 5.1.

Lemma 5.2.

A family of codes with locality containing optimal codes

Authors

Related Research

Cyclic Codes with Locality and Availability

An axiomatization of Λ-quantiles

On bases and the dimensions of twisted centralizer codes

A Study of the Separating Property in Reed-Solomon Codes by Bounding the Minimum Distance

Dynamic Local Searchable Symmetric Encryption

New Constructions of Optimal Cyclic (r,δ) Locally Repairable Codes from Their Zeros

Optimal quaternary (r,δ)-Locally Recoverable Codes: Their Structures and Complete Classification

This week in AI

1 Introduction

2 Quasi affine cartesian codes

Definition 2.1.

Definition 2.2.

Definition 2.3.

Theorem 2.4.

Proof.

3 On the dimension of quasi affine cartesian codes

Definition 3.1.

Definition 3.2.

Proposition 3.3.

Proof.

Corollary 3.4.

Proof.

Definition 3.5.

Theorem 3.6.

Proof.

4 Minimum distance and optimal codes

Theorem 4.1.

Proof.

Corollary 4.2.

Proof.

5 Further results on the minimum distance in a special case

Definition 5.1.

Lemma 5.2.