Storage codes and recoverable systems on lines and grids

Abstract

A storage code is an assignment of symbols to the vertices of a connected graph G(V, E) with the property that the value of each vertex is a function of the values of its neighbors, or more generally, of a certain neighborhood of the vertex in G. In this work we introduce a new construction method of storage codes, enabling one to construct new codes from known ones via an interleaving procedure driven by resolvable designs. We also study storage codes on ${\mathbb Z}$ and ${\mathbb Z}^2$ (lines and grids), finding closed-form expressions for the capacity of several one and two-dimensional systems depending on their recovery set, using connections between storage codes, graphs, anticodes, and difference-avoiding sets.

Linear programming bound

To describe the linear programming bound of Mazumdar et al. [22], let us first define a $\tau$-cover by gadgets on a graph $G = (V, E)$. For a subset of vertices $A\subset G$, let $\textrm{cl}(A):=\{v: {\mathcal {N}}(v) \subseteq A \}$ be the closure of A which contains all vertices in A and their neighbors. A tuple $g = (S_1, S_2, c_1,c_2)$ is called a gadget if

• (1) There exist two sets of vertices $A,B \subseteq V$ such that $S_1 = \textrm{cl}(A) \cup \textrm{cl}(B)$ and $S_2 = \textrm{cl}(A) \cap \textrm{cl}(B)$. We call $S_1$ the outside and $S_2$ the inside of the gadget.

• (2) Colors $c_1$ and $c_2$ are picked from a fixed set of size $\tau$, and they are assigned to all the vertices in $S_1$, $S_2$ respectively. Note that each vertex can have multiple assigned colors.

We call $(\{v\}, \emptyset , c_1,c_2)$ a trivial gadget, and $w(g) = |A| + |B|$ the weight of gadget g. A collection of gadgets forms a $\tau$-cover if, for every color c, all the vertices with color c form a vertex cover. It was shown in [22, Theorem 8] that the total weight of the gadgets that form a $\tau$-cover provides an upper bound on $\textsf{cap}(G)$. More formally, we have the following theorem.

Theorem A.1

(Linear programming (LP) bound [22]) Let $\tau > 0$ be a fixed integer. For each gadget that contains $S \subseteq V$, define $\chi _{g,S} = {\mathbb {1}}\{g\text { is involved in the cover}\}$, where S is a part of g. Thus, each gadget g corresponds to two variables $\chi _{g, S_1}$ and $\chi _{g, S_2}$, where $S_1$ and $S_2$ are the outside and the inside of g,  respectively. Denote by $c_g(S)$ the color of the vertex set S in gadget g.

The capacity $\textsf{cap}(G)$ is bounded above by the solution to the following linear program:

$$\begin{aligned} \text {minimize } \quad&\frac{1}{n\tau } \sum _{g,S} \chi _{g,S} \frac{w(g)}{2} \nonumber \\ \text {s.t., } \quad&\frac{1}{\tau } \sum _{g,S} \chi _{g,S} \frac{w(g)}{2} \in {\mathbb N}\nonumber \\ \quad&\sum _{\begin{array}{c} g,S: u \in S \\ C_g(S) = c \end{array}} \chi _{g,S} + \sum _{\begin{array}{c} g',S': v \in S' \\ C_{g'}(S') = c \end{array}} \chi _{g',S'} \ge 1, \quad \forall (u,v) \in E, \forall c \end{aligned}$$

(12)

$$\begin{aligned}&\chi _{g,S_1} = \chi _{g,S_2}, \quad \forall g \hbox { with outside} S_1 \hbox {and inside} S_2. \end{aligned}$$

(13)

Note that conditions (12) guarantee that the sets form a vertex cover, and (13) states that the inside is involved in the gadget cover if and only if the outside is also involved.

Proof of Proposition 3.7:

Let $g = (S_1, S_2, c_1, c_2)$ be a gadget on G and define $\bar{g} = ({\bar{S}}_1, {\bar{S}}_2,c_1,c_2),$ where $S_1:= \bigcup _{v \in S_1} \bar{v}$ and $S_2:= \bigcup _{v \in S_2} \bar{v}$. If $g_1, \ldots , g_m$ forms a $\tau$-cover of G, then it is straightforward to check that $\bar{g}_1, \ldots , \bar{g}_m$ forms a $\tau$-cover of $\bar{G}$.

Note that $w(\bar{g}) = s w(g)$, thus $\frac{1}{n\tau } w(g) = \frac{1}{ns\tau } w(\bar{g})$, and $\bar{g}_1, \ldots , \bar{g}_m$ induce an upper bound that equals to the bound by $g_1, \ldots , g_m$, which is $\textsf{cap}(G)$ by our assumption. In other words, we have $\textsf{cap}_{q^k}(\bar{G}) \le \textsf{cap}(G)$. The matching lower bound follows by Proposition 3.3.$\square$