Toric codes are a type of evaluation codes introduced by J. P. Hansen in 2000. They are produced by evaluating (a vector space composed by) polynomials at the points of (F∗q)s, the monomials of these polynomials being related to a certain polytope. Toric codes related to hypersimplices are the result of the evaluation of a vector space of square-free homogeneous polynomials of degree d. The dimension and minimum distance of toric codes related to hypersimplices have been determined by Jaramillo et al. in 2021. The next-to-minimal weight in the case d=1 has been determined by Jaramillo–Velez et al. in 2023. In this work, we use tools from Gröbner basis theory to determine the next-to-minimal weight of these codes for d such that 3≤d≤s−22 or s+22≤d<s.

An m-uniform quantum state on n qubits is an entangled state in which every m-qubit subsystem is maximally mixed. Starting with an m-uniform state realized as the graph state associated with an m-regular graph, and a classical [n,k,d≥m+1] binary linear code with certain additional properties, we show that pure [[n,k,m+1]]2 quantum error-correcting codes (QECCs) can be constructed within the codeword stabilized (CWS) code framework. As illustrations, we construct pure [[22r−1,22r−2r−3,3]]2 and [[(24r−1)2,(24r−1)2−32r−7,5]]2 QECCs. We also give measurement-based protocols for encoding into code states and for recovery of logical qubits from code states.

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 to b-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 integers q, n, d, and b ≥ 2, let Ab(n,d,q) denote the largest possible code size for which there exists a q-ary code of length n with a minimum b-symbol distance d. In [3], various upper and lower bounds on Ab(n,d,q) are given for b = 2. In this paper, we generalize some of these bounds to the b-symbol read channels for b > 2 and present several new bounds on Ab(n,d,q). In particular, we establish the linear programming bound, a recurrence relation on Ab(n,d,q), the Johnson bound (even), the restricted Johnson bound, the Gilbert-Varshamov-type bound, and the Elias bound for the metric of symbols b, 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 this note, we study primary monomial affine variety codes defined from the Klein-like curve  x^2y+y^2+x over F_4 . Implementing the techniques suggested by Geil and Özbudak in [3], we estimate the minimum distance of various considered codes. In a few cases, we obtain the exact value of the symbol-pair distance of these codes. Furthermore, we determine lower bounds on the generalized Hamming weights of the codes so obtained. Few codes obtained are the best-known codes according to [5].

Let  Fq be a finite field with q elements, where q is a power of a prime p. A polynomial over F_q  is monomially squarefree if all its monomials are squarefree. In this paper, we determine an upper bound on the number of common zeroes of any set of  linearly independent monomially squarefree polynomials of  Fq[t1,...,ts] in the affine torus T=(Fq^*)^s under certain conditions on r, s  and the degree of these polynomials. Applying the results, we obtain the generalized Hamming weights of toric codes over hypersimplices and squarefree affine evaluation codes, as defined in [14].

A binary linear code is said to be a Z2-double cyclic code if its coordinates can be partitioned into two subsets such that any simultaneous cyclic shift of the coordinates of the subsets leaves the code invariant. These codes were introduced in [6]. A Z2-double cyclic code is called reversible if reversing the order of the coordinates of the two subsets leaves the code invariant. In this note, we give necessary and sufficient conditions for a Z2-double cyclic code to be reversible. We also give a relation between reversible Z2-double cyclic code and LCD Z2-double cyclic code for the separable case and we present a few examples to show that such a relation doesn't hold in the non-separable case. Furthermore, we list examples of reversible Z2-double cyclic codes of length ≤10.

We estimate the minimum distance of primary monomial affine variety codes defined from a hyperelliptic curve  x^5+x-y^2 over F_7 . To estimate the minimum distance of the codes, we apply symbolic computations implementing the techniques suggested by Geil and Özbudak. For some of these codes, we also obtain the symbol-pair distance. Furthermore, lower bounds on the generalized Hamming weights of the constructed codes are obtained. The proposed method to calculate the generalized Hamming weights can be applied to any primary monomial affine variety codes.

Let p be a prime number and s > 0 an integer. In this short note, we investigate one-point geometric Goppa codes associated with an elementary abelian p-extension of F_(p^s)(x). We determine their dimension and exact minimum distance in a few cases. These codes are a special case of weak Castle codes. We also list exact values of the second generalized Hamming weight of these codes in a few cases. Simple criteria for self-duality and quasi-self-duality of these codes are also provided. Furthermore, we construct examples of quantum codes, convolutional codes, and locally recoverable codes on the function field.

Let m_0 and m_1 be two odd positive integers. In this paper, the weight distribution of Z_2-double cyclic codes of length  m_0+m_1 for a special case are determined explicitly.