[Paper Review] Cyclic Codes with Locality and Availability

Lukas Holzbaur, Ragnar Freij-Hollanti+1 more|arXiv (Cornell University)|Dec 17, 2018

Coding theory and cryptography11 references4 citations

TL;DR

This paper presents a construction of cyclic locally repairable codes (LRCs) with strong $(r, ho)$-locality and $t$-availability using a generalized defining set approach based on Tamo-Barg codes. By applying the Hartmann-Tzeng bound and carefully extending the defining set, the authors achieve optimal or near-optimal code dimensions for given minimum distances, particularly under strong orthogonality of repair sets, demonstrating optimality for specific parameter sets.

ABSTRACT

In this work codes with availability are constructed based on the cyclic \emph{locally repairable code} (LRC) construction by Tamo et al. and their extension to $(r,ρ)$-locality by Chen et al. The minimum distance of these codes is increased by carefully extending their defining set. We give a bound on the dimension of LRCs with availability and orthogonal repair sets and show that the given construction is optimal for a range of parameters.

Motivation & Objective

• To design cyclic codes with both $(r, ho)$-locality and multiple disjoint repair sets (availability) for efficient distributed storage.
• To introduce and formalize the concept of strong orthogonality in repair sets, ensuring minimal overlap and maximal independence.
• To derive a dimension bound for LRCs with strong orthogonality and compare it to the constructed codes to assess optimality.
• To improve minimum distance and code dimension by extending the defining set using the Hartmann-Tzeng bound.
• To demonstrate that the proposed construction achieves optimal or near-optimal dimension for a range of parameters, particularly under strong orthogonality.

Proposed method

• Constructs cyclic LRCs based on the Tamo-Barg framework and its generalization to $(r, ho)$-locality via a product structure over finite fields.
• Defines a $t$-dimensional hyperrectangle-based defining set using roots of unity and a generating polynomial $g(x)$ dividing $x^n - 1$.
• Applies the Hartmann-Tzeng bound to derive lower bounds on the minimum distance $d$ by extending the defining set with additional zeros $\mathcal{D}_g$.
• Imposes strong orthogonality on repair sets via a bijection to $\mathbb{Z}/n_1 \times \cdots \times \mathbb{Z}/n_t$, ensuring disjoint repair sets with controlled structure.
• Uses the BCH bound as a special case of the Hartmann-Tzeng bound to estimate minimum distance when $\gamma = 0$.
• Compares the dimension of constructed codes to a theoretical upper bound derived from the dimension constraint under strong orthogonality and repair set parameters.

Experimental results

Research questions

• RQ1 Can cyclic codes be constructed with both $(r, ho)$-locality and $t$-availability while maintaining high minimum distance?
• RQ2 How does strong orthogonality of repair sets affect the achievable code dimension and minimum distance?
• RQ3 Is the proposed construction optimal in terms of dimension for a given minimum distance and locality parameters?
• RQ4 To what extent can the Hartmann-Tzeng bound be leveraged to increase the minimum distance of cyclic LRCs with availability?
• RQ5 What is the theoretical upper bound on the dimension of LRCs with strong orthogonality, and how close does the construction come to it?

Key findings

• The proposed construction achieves optimal dimension for the $[n=15,k=6]$ code over $\mathbb{F}_{16}$ with strong $(\{2,4\},\{2,2\})$-2-availability and minimum distance $d \geq 7$, matching the theoretical bound.
• For the $[n=12,k=4]$ code over $\mathbb{F}_{13}$, the construction achieves $d \geq 8$ using the Hartmann-Tzeng bound, outperforming the previously reported $d \geq 6$.
• In Table 1, several codes (e.g., $n=15, d\geq11, k=4$) achieve the bound from Theorem 3.1, indicating optimality under the given constraints.
• The construction is shown to be optimal when the bound accounts for the floor function in the dimension formula, as demonstrated in Example 1 where $k \leq 6$ is tight.
• The use of $\mathcal{D}_g$ to extend the defining set allows for systematic improvement of the minimum distance while preserving locality and availability properties.
• The bound derived in Theorem 3.1 is shown to be tight for selected parameters, confirming that the construction is optimal or near-optimal in terms of dimension for given $d$, $r$, and $\rho$.