The family of functions plays a central role in the design and effectiveness of function-correcting codes. By focusing on a well-defined family of functions, function-correcting codes can be constructed with minimal length while still ensuring full error detection and correction within that family. In this work, we explore the concept of locally (λ,ρ)-functions forb-symbol read channels and investigate the optimal redundancy of the corresponding function-correctingb-symbol codes (FCBSC) by introducing the notions of locally (λ,ρ,b)-functions. First, we discuss the values ofλandρfor which a function can be considered as a locally (λ,ρ)-function inb-symbol metric. The findings improve some known results in the Hamming metric and present several new results in theb-symbol metric. Then we investigate the optimal redundancy of (f,t)-FCBSCs for locally (λ,ρ,b)-functions. We establish a recurrence relation between the optimal redundancy of (f,t)-function-correcting codes for the (b+ 1)-symbol read andb-symbol read channels. We present an upper bound on the optimal redundancy of (f,t)-function-correctingb-symbol codes for general locally (λ,ρ, b)-functions by associating it to the minimum achievable length ofb-symbol error-correcting codes and traditional Hamming-metric codes, given a fixed number of codewords and a specified minimum distance. We derive some explicit upper bounds on the redundancy of (f,t)-function-correctingb-symbol codes for locally (λ,2t,b)-functions. Moreover, for the case whereb= 1, we show that a locally (3,2t,1)-function achieves the optimal redundancy of 3t. Additionally, we explicitly investigate the locality and optimal redundancy of FCBSCs for theb-symbol weight function and weight distribution function forb≥ 1.

In classical coding theory, error-correcting codes are designed to protect against errors occurring at individual symbol positions in a codeword. However, in practical storage and communication systems, errors often affect multiple adjacent symbols rather than single symbols independently. To address this, symbol-pair read channels were introduced [1], and later generalized tob-symbol read channels [2] to better model such error patterns.b-Symbol read channels generalize symbol-pair read channels to account for clustered errors in modern storage and communication systems. By developing bounds and efficient codes, researchers improve data reliability in applications such as storage devices, wireless networks, and DNA-based storage. Given integersq, n, d,andb≥ 2, letAb(n,d,q) denote the largest possible code size for which there exists aq-ary code of length n with a minimumb-symbol distanced. In [3], various upper and lower bounds onAb(n,d,q) are given forb= 2. In this paper, we generalize some of these bounds to theb-symbol read channels forb> 2 and present several new bounds onAb(n,d,q). In particular, we establish the linear programming bound, a recurrence relation onAb(n,d,q), the Johnson bound (even), the restricted Johnson bound, the Gilbert-Varshamov-type bound, and the Elias bound for the metric of symbolsb,b≥ 2. Furthermore, we provide examples showing that the Gilbert–Varshamov bound established in this paper yields a stronger lower bound than the one given in [4]. Additionally, we introduce an alternative approach to derive the sphere-packing and Plotkin bounds.

In [3], the relation between stabilizer codes and binary codes over the symplectic inner product and symplectic weight was established. In the current work, we present a relation between the symplectic weight and symbol-pair weight and use it to construct stabilizer codes of length n from binary codes of length n defined over the symbol-pair metric and Euclidean inner product. In particular, we use certain length n binary LCD codes over the symbol-pair metric to obtain stabilizer codes of length n. We also present the Modified CSS construction which outperforms the CSS construction in the given setup.

Let p be a prime, and λ be a nonzero element of the finite field Fpm. The λ-constacyclic codes of length psover Fpmare linearly ordered under set-theoretic inclusion, i.e., they are the ideals 〈(x - λ0)i〉, 0 ≤ i ≤ psof the chain ring [(Fpm[x])/((xps- λ))]. This structure is used to establish the symbol-pair distances of all such λ-constacyclic codes. Among others, all maximum distance separable symbol-pair constacyclic codes of length ps are obtained.