pseudo-random array is a two-dimensional array in which eachn1×n2nonzero matrix is contained exactly once as a window in the array. The pseudo-random arrays we consider have many desirable properties in addition, such as the shift-and-add-property, i.e., the addition of the array to any of its nontrivial shifts is another nontrivial shift of the array. A pseudorandom array code is a linear code ofr1×r2arrays in which eachn1×n2nonzero matrix is contained exactly once as a window in one of the arrays. In this paper, new parameters for pseudo-random arrays are presented, and their construction is generalized to pseudo-random array codes. Our constructions of pseudo-random array codes are based on the folding of sequences. Two techniques to verify whether an array or a set of arrays constructed by folding is a pseudo-random array or a pseudo-random array code, respectively, are presented. These verification techniques can also be used for VLSI testing.

BCH codes form an important class of cyclic codes, which have applications in communication and data storage systems. Although the BCH bound provides a lower bound on the minimum distance of BCH codes, determining the true minimum distances of BCH codes is a very challenging problem. In this paper, we settle the minimum distances of a number of infinite families of narrow-sense BCH codes. By explicitly constructing the locator polynomials for minimum weight codewords, we obtain many families of primitive and non-primitive BCH codes withd= δ, wheredis the minimum distance of aq-ary BCH code of lengthn, designed distance δ, and offsetb, denoted by C(q,n,δ,b). For primitive BCH codes, we obtain infinite families of BCH codes over F3and F4satisfyingd= δ, where δ 2 {5, 6, 7, 8}. Moreover, we construct several infinite families ofq-ary BCH codes withd= δ, where 2 ≤ δ ≤q−1. For δ =qt+1, we prove that the BCH code C(q,qm−1,qt+1,1)hasd= δ for allmsatisfyingm≡ 0 (modpt), wherepdenotes the characteristic of Fq. In the paper by Ding et al., IEEE Trans. Inf. Theory 61(5): 2351-2356, it was conjectured that the minimum distance of C(q,qm−1,qt+1,1)is always equal to its Bose distancedB. Our result confirms this conjecture for the casem≡ 0 (modpt). For non-primitive BCH codes, we construct a family of BCH codes C(q, qp−1/λ ,p+1,1)withd= δ =p+ 1, wherepis an odd prime,q = pewithp∤eand λ |q− 1.

MDS codes with determined Hermitian hull dimensions have attracted significant attention for their application in quantum error correction. From an MDS code over Fq2with fixed Hermitian hull dimension ℓ, whereqis a prime power larger than 2, one can obtain an MDS code with any smaller ℓ′-dimensional Hermitian hull for 0 ≤ ℓ′ ≤ ℓ. Then it is natural to consider the problem of determining the maximum Hermitian hull dimension, denoted byLq(n, k), among all MDS codes with the same lengthnand dimensionkover Fq2. Some constructions of Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes had been proposed, which addressed this problem for certain parameter regimes. However, it is still unknown for many cases, in particular fork≥q+ 1. In this paper, we study the Hermitian hulls of a class of codes which generalizes GRS codes, called twisted generalized Reed-Solomon (TGRS) codes. TGRS codes contain MDS subclasses that are not linearly equivalent to GRS codes (called non-GRS codes). We give a bound on the Hermitian hull dimensions of certain TGRS codes of general twists. In addition, we derive a lower bound onLq(n, k) forn|q2− 1 and 1 ≤k≤n, which generalizes and improves some previous results. For some parameter regimes wheren≥q+ 1 andk≥q+ 1, we prove thatLq(n, k) ≥k/2 and explicitly construct [n, k]q2MDS codes whose Hermitian hulls have dimension at leastk/2. This result solves partially an open problem pointed out in the literature. The constructed MDS codes arise from either GRS or non-GRS TGRS codes. Furthermore, some sufficient conditions for TGRS codes with general twists to be Hermitian self-orthogonal are given, and Hermitian self-orthogonal non-GRS MDS codes are constructed. Based on our constructions, we provide several families of MDS entanglement-assisted quantum error-correcting codes.

Sum-rank codes have wide applications in multishot network coding, distributed storage and the construction of space-time codes. Asymptotically good sequences of linearized algebraic geometry sum-rank codes, exceeding the Gilbert-Varshamov-like bound, were constructed in a recent paper published in IEEE Trans. Inf. Theory by E. Berardini and X. Caruso. We call this bound the Tsfasman-Vlăduţ-Zink-like bound. In this paper, we introduce the concatenation of a sum-rank code and a Hamming metric code. Then many sum-rank codes with good parameters, which are better than sum-rank BCH codes, are constructed simply and explicitly. Moreover, we obtain an asymptotically good sequence of sum-rank codes exceeding the Tsfasman-Vlăduţ-Zink-like bound and the Gilbert-Varshamov-like bound.

In a recent work, quantum locally recoverable codes (qLRCs) have been introduced for their potential application in large-scale quantum data storage and implication for quantum LDPC codes. This work focuses on the bounds and constructions of qLRCs derived from the Hermitian construction, which solves an open problem proposed by Luo $et~al.$ (IEEE Trans. Inf. Theory, 71 (3): 1794-1802, 2025). We present four bounds for qLRCs and give comparisons in terms of their asymptotic formulas. We construct several new infinite families of NMDS codes, with general and flexible dimensions, that support t-designs for $t\in \{2,3\}$, and apply them to obtain Hermitian dual-containing classical LRCs (cLRCs). As a result, we derive three explicit families of optimal qLRCs. Compared to the known qLRCs obtained by the CSS construction, our optimal qLRCs offer new and more flexible parameters. It is also worth noting that the constructed cLRCs themselves are interesting as they are optimal with respect to four distinct bounds for cLRCs.

Pseudo-random arrays are the two-dimensional analog of M-sequences. Pseudo-random array codes are the twodimensional analog of sequences generated by a product of irreducible polynomials with the same exponent. The union of the arrays in such a code has the window property and the shift-and-add property, implying that these codes are linear. The folding technique is the most basic one for forming such arrays and codes. A new criterion for generating pseudo-random arrays based on folding is given. This new criterion yields pseudo-random arrays with new parameters. A general construction for such array codes is given. It appears that the arrays generated in this construction can be constructed by folding the nonzero sequences generated by a product of irreducible polynomials of the same degree and the same exponent. Two hierarchies of the pseudo-random array codes are provided. In one hierarchy codewords of one code with smaller windows are contained in codewords of another code which stands above him in the hierarchy. The second hierarchy is a partition of the pseudo-random array codes generated by folding into classes based on the polynomial types which participate in their construction.

In eight published papers in IEEE Transactions on Information Theory, infinite families of optimal few-weight binary andq-ary linear codes were constructed and their weight distributions were determined. These codes are linear codes meeting the Griesmer bound. We indicate that many Griesmer codes constructed in these papers are not new. They are actually Solomon-Stiffler codes invented in 1965. Therefore weight distributions of some special binary orq-ary Solomon-Stiffler codes were determined in the papers mentioned above. From a similar geometric approach as Solomon-Stiffler codes, we construct ten infinite families of binary, ternary and quaternary few-weight, optimal, almost optimal and near-optimal linear codes close to the Griesmer bound and their weight distributions are determined. These linear codes have positive Griesmer defects up to five, and thus not Solomon-Stiffler codes and Griesmer codes from minihypers. Moreover, many optimal, best known and almost optimal quantum codes of small lengths, comparing with Grassl's table on quantum codes, are constructed from the same geometric approach as binary Solomon-Stiffler codes.

Window Weight Limited (WWL) strings and codes have been studied recently owing to their various applications, including DNA based storage and energy-harvesting. In this work, we proposed to study a new coding scheme, called WWL Gray code. This is a list of WWL strings in Gray order, that is, the Hamming distance of two consecutive strings in the list is one. We are the first to investigate the WWL Gray codes with efficient rankinglunranking algorithms. We show that the WWL Gray codes are useful in designing robust positioning sequences. Note that the robust positioning sequences have attracted a lot of attentions recently owing to their numerous applications, including quantum communications and robot localization. In this work, using the WWL Gray codes, we obtain robust positioning sequences with less redundancy compared to the best known results.

Sum-rank-metric codes have attracted lots of attention due to their numerous applications, including muti-shot linear network coding, space-time coding, and distributed storage systems. In this paper, we focus on constructing linear sum-rank-metric codes achieving Singleton bound, which are called maximum sum-rank distance (MSRD) codes. This family of codes is the analogue of maximum distance separable (MDS) codes in Hamming metric. We propose two constructions of linear MSRD codes with various matrix sizes. Each of them yields new MSRD codes with different parameter regimes, and one of them generalizes some recent results of Byrne et al. (2021) and Chen (2023). The block lengths and the matrix block sizes of our codes are not restricted to the sizes of the finite field. Our technique is mainly based on rank metric codes and their sub-codes of different minimum rank distances.

One of the main problems of subspace coding is to determine the maximal size of a constant dimension subspace code with given parameters. In this paper, we show that mixed dimension subspace codes can be used to construct large constant dimension subspace codes. We introduce a new class of subspace codes called mixed dimension/distance subspace codes. Using such codes, we present two constructions for large constant dimension subspace codes. The problem about the sizes of our constant dimension subspace codes is transformed into finding mixed dimension/distance subspace codes with large dimension distributions. The new constructed codes are the largest known for many sets of parameters. Our method gives at least 136 new lower bounds on the sizes of constant dimension subspace codes.