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Abstract The Generalized Hamming weights and their relative version, which generalize the minimum distance of a linear code, are relevant to numerous applications, including coding on the wire-tap channel of type II,t-resilient functions, bounding the cardinality of the output in list decoding algorithms, ramp secret sharing schemes, and quantum error correction. The generalized Hamming weights have been determined for some families of codes, including Cartesian codes and Hermitian one-point codes. In this paper, we determine the generalized Hamming weights of decreasing norm-trace codes, which are linear codes defined by evaluating sets of monomials that are closed under divisibility on the rational points of the extended norm-trace curve given by$$x^{u} = y^{q^{s - 1}} + y^{q^{s - 2}} + \cdots + y$$ $x^u=y^{q^{s-1}}+y^{q^{s-2}}+\cdots +y$over the finite field of cardinality$$q^s$$ $q^s$, whereuis a positive divisor of$$\frac{q^s - 1}{q - 1}$$ $\frac{q^s-1}{q-1}$. As a particular case, we obtain the weight hierarchy of one-point norm-trace codes and recover the result of Barbero and Munuera (2001) giving the weight hierarchy of one-point Hermitian codes. We also study the relative generalized Hamming weights for these codes and use them to construct impure quantum codes with excellent parameters. 

Abstract Codes with locality, also known as locally recoverable codes, allow for recovery of erasures using proper subsets of other coordinates. These subsets are typically of small cardinality to promote recovery using limited network traffic and other resources. Hierarchical locally recoverable codes allow for recovery of erasures using sets of other symbols whose sizes increase as needed to allow for recovery of more symbols. In this paper, we describe a hierarchical recovery structure arising from geometry in Reed–Muller codes and codes with availability from fiber products of curves. We demonstrate how the fiber product hierarchical codes can be viewed as punctured subcodes of Reed–Muller codes, uniting the two constructions. This point of view provides natural structures for local recovery with availability at each level in the hierarchy. 

Abstract In this paper, we introduce curve-lifted codes over fields of arbitrary characteristic, inspired by Hermitian-lifted codes over$$\mathbb {F}_{2^r}$$ $F_{2^r}$. These codes are designed for locality and availability, and their particular parameters depend on the choice of curve and its properties. Due to the construction, the numbers of rational points of intersection between curves and lines play a key role. To demonstrate that and generate new families of locally recoverable codes (LRCs) with high availabilty, we focus on norm-trace-lifted codes. 

Abstract CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes$$(C_1, C_2)$$ $(C_1,C_2)$such that$$C_1$$ $C_1$contains$$C_2$$ $C_2$,$$C_2$$ $C_2$is even, and the shortening of the dual of$$C_1$$ $C_1$with respect to the support of each codeword of$$C_2$$ $C_2$is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes$$(C_1, C_2)$$ $(C_1,C_2)$is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature. 

In this paper, we present a linear-time decoding algorithm for expander codes based on irregular graphs of any positive vertex expansion factor [Formula: see text] and inner codes with a minimum distance of at least [Formula: see text], where [Formula: see text] is the maximum right degree. The algorithm corrects a constant fraction of errors. It builds on two thrusts. The first is a series of works starting with that of Sipser and Spielman [Expander codes, IEEE Trans. Inf. Theory 42(6) (1996) 1710–1722] demonstrating that an asymptotically good family of error-correcting codes that can be decoded in linear time even from a constant fraction of errors in a received word provided [Formula: see text] is at least [Formula: see text] and continuing to the results of Gao and Dowling [Fast decoding of expander codes, IEEE Trans. Inf. Theory 64(2) (2018) 972–978], which only require [Formula: see text] provided the inner code minimum distance is sufficiently large. The second is the improved performance of LDPC codes based on irregular graphs demonstrated by Luby et al. [Improved low- density parity-check codes using irregular graphs, IEEE Trans. Inf. Theory 47(2) (2001) 585–598] and Richardson et al. [Design of capacity- approaching irregular low-density parity-check codes, IEEE Trans. Inf. Theory 47(2) (2001) 619–637]. The algorithm presented in this paper allows for irregular or regular graph-based constructions and uses inner codes of appropriate lengths as checks rather than simple parity-checks. 

The ability of a linear error-correcting code to recover erasures is connected to influences of particular monotone Boolean functions. These functions provide insight into the role that particular coordinates play in a code’s erasure repair capability. We consider directly the influences of coordinates of a code. We describe a family of codes, called codes with minimum disjoint support, for which all influences may be determined. As a consequence, we find influences of repetition codes and certain distinct weight codes. Computing influences is typically circumvented by appealing to the transitivity of the automorphism group of the code. Some of the codes considered here fail to meet the transitivity conditions required for these standard approaches, yet we can compute them directly.