Function-Correcting Codes for b-Symbol Read Channels

TL;DR

This paper generalizes function-correcting codes to $b$-symbol read channels over finite fields by introducing irregular $b$-symbol distance codes and a graphical independent-set framework. It establishes precise relationships between function-correcting codes and irregular-distance codes, deriving bounds on the optimal redundancy $r_b^f(k,t)$ via matrices $\boldsymbol{B}_f^{(1)}(t)$ and $\boldsymbol{B}_f^{(2)}(t)$, and shows equivalence to independent sets in the function-dependent graph $G_f^b(k,t,r)$. It also provides concrete constructions and bounds for special function classes, including $b$-symbol locally binary functions, $wt_b$, and $wt_b$-distribution functions, showing potential redundancy reductions compared to classical $b$-symbol codes. The results have practical relevance for modern storage systems with $b$-symbol reads, enabling more compact yet reliable protection of function evaluations. Overall, the work blends graph-based methods, irregular-distance coding theory, and function-specific encodings to advance function-correcting codes in the broader $b$-symbol channel setting.

Abstract

Function-correcting codes are an innovative class of codes that are designed to protect a function evaluation of the data against errors or corruptions. Due to its usefulness in machine learning applications and archival data storage, where preserving the integrity of computation is crucial, Lenz et al. recently introduced function-correcting codes for binary symmetric channels to safeguard function evaluation against errors. Xia et al. expanded this concept to symbol-pair read channels over binary fields. The current paper further advances the theory by developing function-correcting codes for b-symbol read channels over finite fields. We introduce the idea of irregular b-symbol distance codes and establish bounds on their performance over finite fields. This concept helps in understanding the behavior of function-correcting codes in more complex settings. We also present a graphical approach of the problem of constructing function-correcting b-symbol codes. Furthermore, we apply these general concepts to specific classes of functions and compare the redundancy of function-correcting b-symbol codes with classical b-symbol codes. Our findings demonstrate that function-correcting b-symbol codes achieve lower redundancy while maintaining reliability.

Figures (4)

• Figure 1: Setting for function correction in $b-$symbol read channel.
• Figure 2: Graph $G_f^b(k,t,r)$ for $b = 3, k = 4, t = 2, r = 2$ and function $f((x_1,x_2,x_3,x_4)) = x_1 \vee x_2 \vee x_3 \vee x_4$ over binary field. The nodes in red boxes that are 16 in size form a function-correcting $b-$symbol code for function $f$
• Figure 3: Graph $G_{wt_3}^3(k,t,r)$ for $k = 4, t = 2, r =2$. The independent set of size 16 is highlighted in red boxes, forming the function-correcting $b-$symbol codes for $wt_3$ function.
• Figure 4: The figure illustrates the layouts of $(k,l, L_j)$ subsets for $L_j = 4$ and $j = 1$, where the lighter coloured rectangles represent unoccupied boxes, darker rectangles represent occupied rectangles and orange represents size 1 unoccupied subset. Category 1 depicts layouts where neither the occupied nor the unoccupied rectangles wrap around the ends. Category 2 represents cases where an occupied rectangle wraps around the ends and Category 3 shows the case when an unoccupied rectangle wraps around.

Theorems & Definitions (63)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• Example 2.4
• Proposition 2.5
• Lemma 2.6