Generalization of the Fano and Non-Fano Index Coding Instances

TL;DR

This work investigates how linear index coding rates depend on the underlying field characteristic by leveraging Fano and non-Fano matroids and their field-parameterized $p$-variants. The authors provide a completely matrix-based, independent proof of the characteristic dependence, reintroduce the $p$-Fano and $p$-non-Fano matroid instances, and show how these lead to two new index coding classes with size polynomial in $p$. Specifically, the $p$-Fano index coding instances achieve optimal linear rates only over fields of characteristic $p$, while the $p$-non-Fano instances are optimally linear over any characteristic other than $p$. The constructions yield compact index coding problems with a clear separation between characteristic-dependent linear codes and general non-linear capabilities, advancing understanding of when linear strategies suffice and offering practical encodings with reduced instance sizes. Overall, the paper deepens the link between matroid representations and index coding, providing concrete, efficient instances and techniques for analyzing linear coding limits across field characteristics.

Abstract

Matroid theory is fundamentally connected with index coding and network coding problems. In fact, the reliance of linear index coding and network coding rates on the characteristic of a field has been demonstrated by using the two well-known matroid instances, namely the Fano and non-Fano matroids. This established the insufficiency of linear coding, one of the fundamental theorems in both index coding and network coding. While the Fano matroid is linearly representable only over fields with characteristic two, the non-Fano instance is linearly representable only over fields with odd characteristic. For fields with arbitrary characteristic $p$, the Fano and non-Fano matroids were extended to new classes of matroid instances whose linear representations are dependent on fields with characteristic $p$. However, these matroids have not been well appreciated nor cited in the fields of network coding and index coding. In this paper, we first reintroduce these matroids in a more structured way. Then, we provide a completely independent alternative proof with the main advantage of using only matrix manipulation rather than complex concepts in number theory and matroid theory. In this new proof, it is shown that while the class $p$-Fano matroid instances are linearly representable only over fields with characteristic $p$, the class $p$-non-Fano instances are representable over fields with any characteristic other than characteristic $p$. Finally, following the properties of the class $p$-Fano and $p$-non-Fano matroid instances, we characterize two new classes of index coding instances, respectively, referred to as the class $p$-Fano and $p$-non-Fano index coding, each with a size of $p^2 + 4p + 3$.

Figures (4)

• Figure 1: $\boldsymbol{H}_{p}\in \mathbb{F}_{q}^{(p+1)\times n}$: If $\mathbb{F}_{q}$ has characteristic $p$, then $\boldsymbol{H}_{p}$ is a scalar linear representation of the class $p$-Fano matroid $\mathcal{N}_{F}(p)$, and if $\mathbb{F}_{q}$ has any characteristic other than characteristic $p$, then $\boldsymbol{H}_{p}$ is a scalar linear representation of the class $p$-non-Fano matroid $\mathcal{N}_{nF}(p)$.
• Figure 2: $\boldsymbol{H}_{p}\in \mathbb{F}_{q}^{(p+1)\times m}$: If $\mathbb{F}_{q}$ does have characteristic $p$, then $\boldsymbol{H}_{p}$ is an encoding matrix for the class $p$-Fano index coding instance $\mathcal{I}_{F}(p)$, and if $\mathbb{F}_{q}$ does have any characteristic other than characteristic $p$, then $\boldsymbol{H}_{p}$ is an encoding matrix for the class $p$-non-Fano index coding instance $\mathcal{I}_{nF}(p)$.
• Figure 3: $\boldsymbol{H}_{p=2}\in \mathbb{F}_{q}^{3\times 19}$: If $\mathbb{F}_{q}$ does have characteristic two (such as $GF(2)$), then $\boldsymbol{H}_{2}$ is an encoding matrix for the index coding instance $\mathcal{I}_{F}(2)$, and if $\mathbb{F}_{q}$ does have odd characteristic any characteristic other than characteristic two such as $GF(3)$), then $\boldsymbol{H}_{2}$ is an encoding matrix for the index coding instance $\mathcal{I}_{nF}(2)$.
• Figure 4: $\boldsymbol{H}_{p=3}\in \mathbb{F}_{q}^{4\times 33}$: If $\mathbb{F}_{q}$ does have characteristic three (such as $GF(3)$), then $\boldsymbol{H}_{3}$ is an encoding matrix for the index coding instance $\mathcal{I}_{F}(3)$, and if $\mathbb{F}_{q}$ does have any characteristic other than characteristic three (such as $GF(2)$), then $\boldsymbol{H}_{3}$ is an encoding matrix for the index coding instance $\mathcal{I}_{nF}(3)$.

Theorems & Definitions (70)

• Definition 1: $\mathcal{C}_{\mathcal{I}}$: Index Code for $\mathcal{I}$
• Definition 2: $\beta(\mathcal{C}_{\mathcal{I}})$: Broadcast Rate of $\mathcal{C}_{\mathcal{I}}$
• Definition 3: $\beta(\mathcal{I})$: Broadcast Rate of $\mathcal{I}$
• Definition 4: Linear Index Code
• Proposition 1: Arman-3-2
• Definition 5: $\lambda_{q}(\mathcal{L}_{\mathcal{I}})$: Linear Broadcast Rate of $\mathcal{L}_{\mathcal{I}}$ over $\mathbb{F}_q$
• Definition 6: $\lambda_{q}(\mathcal{I})$: Linear Broadcast Rate of $\mathcal{I}$ over $\mathbb{F}_q$
• Definition 7: $\lambda(\mathcal{I})$: Linear Broadcast Rate for $\mathcal{I}$
• Definition 8: Scalar and Vector Linear Index Code
• Definition 9: Independent Set of $\mathcal{I}$