Matroid Theory and Storage Codes: Bounds and Constructions

Abstract

Recent research on distributed storage systems (DSSs) has revealed interesting connections between matroid theory and locally repairable codes (LRCs). The goal of this chapter is to introduce the reader to matroids and polymatroids, and illustrate their relation to distributed storage systems. While many of the results are rather technical in nature, effort is made to increase accessibility via simple examples. The chapter embeds all the essential features of LRCs, namely locality, availability, and hierarchy alongside with related generalised Singleton bounds.

The authors gratefully acknowledge the financial support from the Academy of Finland (grants #276031 and #303819), as well as the support from the COST Action IC1104.

Appendix: More About Matroid Theory

A matroid realisation of an $\mathbb {F}$-linear matroid M has two geometric interpretations. Firstly, we may think of a matrix representing M as a collection of n column vectors in $\mathbb {F}^k$. As the matroid structure is invariant under row operations, or in other words under change of basis in $\mathbb {F}^k$, we tend to think of M as a configuration of n points in abstract projective k-space.

The second interpretation comes from studying the row space of the matrix, as an embedding of $\mathbb {F}^k$ into $\mathbb {F}^n$. Row operations correspond to a change of basis in $\mathbb {F}^k$, and hence every matroid representation can be thought of as a k-dimensional subspace of $\mathbb {F}^n$. In other words, a matroid representation is a point in the Grassmannian $\mathrm {Gr}(n,k; \mathbb {F})$, and $\mathrm {Gr}(n,k; \mathbb {F})$ has a stratification as a union of realisation spaces R(M), where M ranges over all $\mathbb {F}$-representable matroids of size n and rank k. This perspective allows a matroidal perspective also on the subspace codes discussed in chapters “Codes Endowed with the Rank Metric”–“Generalizing Subspace Codes to Flag Codes Using Group Actions”, where the codewords themselves are matroid representations. However, so far this perspective has not brought any new insights to the topic.

Another instance where matroids appear naturally in mathematics is graph theory. Let $\varGamma$ be a finite graph with edge set E. We obtain a matroid $M_\varGamma =(\mathscr {I}, E)$, where $I\subseteq E$ is independent if the subgraph $\varGamma _I\subseteq \varGamma$ induced on $I\subseteq E$ is a forest, i.e., has no cycles. A matroid that is isomorphic to $M_\varGamma$ for some graph $\varGamma$ is said to be a graphical matroid.

Example 15

The matrix G and the graph $\varGamma$ given below generate the same matroid, regardless of the field over which G is defined.

Some examples of independent sets in G and $\varGamma$ are $\{3,4,6\}, \{1,2,3,5\}, \{2,3,4,6\}$. The set $X = \{5,6,7\}$ is dependent in $M_\varGamma$ as these edges form a cycle, and it is dependent in $M_G$ as the submatrix

has linearly dependent columns.

Indeed, graphical matroids are representable over any field $\mathbb {F}$. To see this, for a graph $\varGamma$ with edge set E, we will construct a matrix $G(\varGamma )$ over $\mathbb {F}$ with column set E as follows. Choose an arbitrary spanning forest $T\subseteq E$ in $\varGamma$, and index the rows of $G(\varGamma )$ by T. Thus $G(\varGamma )$ is a $T\times E$-matrix. Choose an arbitrary orientation for each edge in the graph. For $e\in T\subseteq E$ and $uv\in E$, the entry in position $(e,\{uv\})$ is 1 (respectively $-1$) if e is traversed forward (respectively backward) in the unique path from u to v in the spanning forest T. In particular, the submatrix $G(\varGamma )(T)$ is an identity matrix. It is straightforward to check that the independent sets in $G(\varGamma )$ are exactly the noncyclic sets in $\varGamma$.

Example 16

The matrix G in Example 15 is $G(\varGamma )$ where $\varGamma$ is the graph in the same example, and the spanning forest T is chosen to be $\{1,2,3,4\}$.

The restriction to $X\subseteq E$ of a graphical matroid $M_\varGamma$ is obtained by the subgraph of $\varGamma$ containing precisely the edges in X.

A third example of matroids occurring naturally in mathematics are algebraic matroids [22]. These are associated to field extensions $\mathbb {F}:K$ together with a finite point sets $E\subseteq K$, where the independent sets are those $I\subseteq E$ that are algebraically independent over $\mathbb {F}$. In particular, elements that are algebraic over $\mathbb {F}$ have rank zero, and in general $\rho (I)$ is the transcendence degree of the field extension $\mathbb {F}(I):\mathbb {F}$.

It is rather easy to see that every $\mathbb {F}$-linear matroid is also algebraic over $\mathbb {F}$. Indeed, let $X_1,\ldots , X_k$ be indeterminates, and let

$$g:\mathbb {F}^k\rightarrow \mathbb {F}(X_1,\ldots , X_k)$$

be given by $\mathrm {e}_i\mapsto X_i$ for $i=1,\ldots ,k$. Then $J\subseteq E$ is linearly independent over $\mathbb {F}$ if and only if $\{g(j):j\in J\}$ is algebraically independent over $\mathbb {F}$. Over fields of characteristic zero the converse also holds, so that all algebraic matroids have a linear representation. However, in positive characteristic there exist algebraic matroids that are not linearly representable. For example, the non-Pappus matroid of Example 4 is algebraically representable over $\mathbb {F}_4$, although it is not linearly representable over any field [21]. The smallest example of a matroid that is not algebraic over any field is the Vamos matroid, in Fig. 2 [15].

Definition 21

The dual of $M = (\rho ,E)$ is $M^*=(\rho ^*,E)$, where

$$ \rho ^*(X)=|X|+\rho (E\setminus X) -\rho (E). $$

The definition of the dual matroid lies in the heart of matroid theory, and has profound interpretations. In geometric terms, let M be represented by a k-dimensional subspace V of $\mathbb {F}^n$. Then, the matroid dual $M^*$ is represented by the orthogonal complement $V^\perp \subseteq \mathbb {F}^n$. Surprisingly and seemingly unrelatedly, if $\varGamma$ is a planar graph and $M=M_\varGamma$ is a graphical matroid, then $M^*=M_{\bar{\varGamma }}$, where $\bar{\varGamma }$ is the planar dual of $\varGamma$. Moreover, the dual $M_\varGamma ^*$ of a graphical matroid is graphical if and only if $\varGamma$ is planar.

Fig. 2

The Vamos matroid of size 8 and rank 4, which is not algebraically representable

Definition 22

The contraction of $X\subseteq E$ in the matroid $M = (\rho ,E)$ is $M/X=(\rho ',M\setminus X)$, where $\rho '(Y)=\rho (Y\cup X)-\rho (X)$.

Contraction is the dual operation of deletion, in the sense that $M/X=(M^*_{|E\setminus X})^*$. The terminology comes from graphical matroids, where contraction of the edge $e\in E$ corresponds to deleting e and identifying its endpoints in the graph. Notice that it follows directly from submodularity of the rank function that $\rho _{M/X}(Y)\le \rho _{M|_{E\setminus X}}(Y)$ for every $Y\subseteq E\setminus X$. In terms of subspace representations, contraction of $e\in E$ corresponds to intersecting the subspace that represents M with the hyperplane $\{x_e=0\}$.

As matroids are used as an abstraction for linear codes, it would be desirable to have a way to go back from matroids to codes, namely to determine whether a given matroid is representable, and when it is, to find such a representation. Unfortunately, there is no simple criterion to determine representability [25, 43]. However, there are a plethora of sufficient criteria to prove nonrepresentability, both over a given field and over fields in general. In recent years, these methods have been used to prove two long-standing conjectures, that we will discuss in sections Rota’s Conjecture and Most Matroids are Nonrepresentable respectively.

1.1 Rota’s Conjecture

While there is no simple criterion to determine linear representability, the situation is much more promising if we consider representations over a fixed field. It has been known since 1958, that there is a simple criterion for when a matroid is binary representable.

Theorem 11

([42]) Let $M = (\rho ,E)$ be a matroid. The following two conditions are equivalent.

• 1. M is linearly representable over $\mathbb {F}_2$.

• 2. There are no sets $X\subseteq Y\subseteq E$ such that M|Y / X is isomorphic to the uniform matroid $U_4^2$.

In essence, this means that the only obstruction that needs to be overcome in order to be representable over the binary alphabet, is that no more than three nonzero points can fit in the same plane. For further reference, we say that a minor of the matroid $M = (\rho ,E)$ is a matroid of the form M|Y / X, for $X\subseteq Y\subseteq E$. Clearly, if M is representable over $\mathbb {F}$, then so is all its minors. Let $L(\mathbb {F})$ be the class of matroids that are not representable over $\mathbb {F}$, but such that all of their minors are $\mathbb {F}$-representable. Then the class of $\mathbb {F}$-representable matroids can be written as the class of matroids that does not contain any matroid from $L(\mathbb {F})$ as a minor. Gian-Carlo Rota conjectured in 1970 that $L(\mathbb {F})$ is a finite set for all finite fields $\mathbb {F}$. A proof of this conjecture was announced by Geelen, Gerards and Whittle in 2014, but the details of the proof still remain to written up [12].

Theorem 12

For any finite field $\mathbb {F}$, there is a finite set $L(\mathbb {F})$ of matroids such that any matroid M is representable if and only if it contains no element from $L(\mathbb {F})$ as a minor.

Since the 1970s, it has been known that a matroid is representable over $\mathbb {F}_3$ if and only if it avoids the uniform matroids $U_5^2$, $U_5^3$, the Fano plane $P^2(\mathbb {F}_2)$, and its dual $P^2(\mathbb {F}_2)^*$ as minors. The list $L(\mathbb {F}_4)$ has seven elements, and was given explicitly in 2000. For larger fields, the explicit list is not known, and there is little hope to even find useful bounds on its size.

1.2 Most Matroids are Nonrepresentable

For a fixed finite field $\mathbb {F}$, it follows rather immediately from the minor-avoiding description in the last section that the fraction of n-symbol matroids that is $\mathbb {F}$-representable goes to zero as $n\rightarrow \infty$. It has long been a folklore conjecture that this is true even when representations over arbitrary fields are allowed. However, it was only in 2016 that a verifiable proof of this claim was announced [26].

Theorem 13

$$ \lim _{n \rightarrow \infty } \frac{\# \text {linear matroids on } n \text { elements}}{\# \text {matroids on } n \text { elements }} = 0. $$

The proof is via estimates of the denominator and enumerator of the expression in Theorem 13 separately. Indeed, it is shown in [19] that the number of matroids on n nodes is at least $\varOmega (2^{(2-\varepsilon )^n})$ for every $\varepsilon >0$. The proof of Theorem 13 thus boiled down to proving that the number of representable matroids is $O(2^{n^3})$. This is in turn achieved by bounding the number of so called zero-patterns of polynomials.

1.3 Gammoid Construction of Singleton-Optimal LRCs

For completeness, we end this appendix with a theorem that explicitly presents the matroids constructed in Theorem 7 as gammoids. As discussed in Sect. 4.2, this proves the existence of Singleton-optimal linear LRCs whenever a set system satisfying (8) exists.

Theorem 14

([47], $M(F_1,\ldots ,F_m,E;k;\rho )$-matroids are gammoids) Let $M(F_1,\ldots ,F_m;\rho )$ be a matroid given by Theorem 7 and define $s: E \rightarrow 2^{[m]}$ where $s(x) = \{i \in [m] : x \in F_i\}$. Then $M(F_1,\ldots ,F_m,E;k;\rho )$ is equal to the gammoid $M(\varGamma ,E,T)$, where $\varGamma = (V,D)$ is the directed graph with

$$ \begin{array}{rl} (i) &{} V = E \cup H \cup T \text { where } E,H,T \text { are pairwise disjoint},\\ (ii) &{} T = [k],\\ (iii) &{} H \text { equals the union of the pairwise disjoint sets}\\ &{} H_1, \ldots ,H_m, H_{\ge 2}, \text { where }\\ &{} \begin{array}{l} |H_i| = \rho (F_i) - |\{x \in F_i : |s(x)| \ge 2\}| \text { for } i \in [m],\\ H_{\ge 2} = \{h_y : y \in E \hbox {, }|s(y)| \ge 2 \},\\ \end{array}\\ (iv) &{} D = D_1 \cup D_2 \cup D_3 \text {, where}\\ &{} \begin{array}{l} D_1 = \bigcup _{i \in [m]}\{(\overrightarrow{x,y}) : x \in E \hbox {, }s(x) = \{i\} \hbox {, }y \in H_i\},\\ D_2 = \{(\overrightarrow{x,h_y}) : x \in E \hbox {, }h_y \in H_{\ge 2} \hbox {, }s(x) \subseteq s(y) \},\\ D_3 = \{(\overrightarrow{x,y}) : x \in H, y \in T\}.\\ \end{array} \end{array} $$