In this paper, we study algebraic geometry (AG) codes with respect to the Lee metric. We determine a new lower bound on the minimum Lee distance of AG codes. Using the AG codes, we construct non-linear binary codes that can correct deletions and insertions of zero-symbols.

Motivated by the Hermitian construction of quantum codes using classical linear codes as ingredients, we study the Hermitian hulls of two-point rational algebraic geometry (AG) codes Cℒ(D,G) with G = (k−1)O+rP for some positive integers k and r. We provide a lower bound on their Hermitian hull dimensions. Specifically, we prove that under certain conditions, the Hermitian hull dimensions of such AG codes are lower bounded by k−1−r, giving rise to linear codes of Hermitian hull dimension k−r−1 by some propagation rules. As an application, we derive two families of entanglement-assisted quantum error-correcting codes with new parameters.

We craft a special class of unitary matrices over$\mathbb{F}q^{2}$to generate classical linear codes whose Hermitian hulls have dimensions that we can design. We then use the codes as classical ingredients in the construction of good entanglement-assisted quantum codes. Over finite fields of characteristic 2, we propose two explicit constructions. To highlight their efficacy, we list excellent qubit codes whose parameters are either new or strictly better than comparable best-known codes in the literature.

We study the Hermitian hulls of two-point algebraic geometry codes. Under specific conditions on the Weil differential form associated with the Hermitian dual code, we explicitly determine the hull dimension. We construct k-dimensional linear codes, whose Hermitian hulls have dimension$k-2$, from some algebraic plane curves.

We study the Hermitian hulls of matrix product codes and use them to design linear codes with arbitrary hull dimensions. We continue by looking into some propagation rules that preserve the hull dimensions of$\mathbb{F}_{4}$-linear codes. We propose a recursive method to keep the Hermitian hull dimension fixed while increasing the dimensions or the minimum distances of codes built from a given$\mathbb{F}_{4}$-linear code. Using the Hermitian construction route, we derive good entanglement-assisted quantum codes from matrix product codes and recursively iterated codes.

Hermitian hulls of linear codes are interesting for theoretical and practical reasons alike. In terms of recent application, linear codes whose hulls meet certain conditions have been utilized as ingredients to construct entanglement-assisted quantum error correcting codes. This family of quantum codes is often seen as a generalization of quantum stabilizer codes. Theoretically, compared with the Euclidean setup, the Hermitian case is much harder to deal with. Hermitian hulls of MDS linear codes with low dimensions have been explored, mostly from generalized Reed-Solomon codes. Characterizing Hermitian hulls which themselves are MDS appears to be more involved and has not been extensively studied. This paper introduces some tools to study linear codes whose Hermitian hulls are MDS. Using the tools, we then propose explicit constructions of such codes. We consider Hermitian hulls of both Reed-Solomon and non Reed-Solomon types of linear MDS codes. We demonstrate that, given the same Hermitian hull dimensions, the codes from our constructions have dimensions which are larger than those in the literature.

The Euclidean hull of a linear code$C$is the intersection of$C$with its Euclidean dual$C^\perp $. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correcting codes. In this paper, we firstly consider linear codes with one-dimensional Euclidean hull from algebraic geometry codes, and then present a general method to construct linear codes with arbitrary dimensional Euclidean hull. Some new EAQECCs are presented.

In this paper, we study self-dual codes over finite fields using tools from algebraic function fields in one variable. An algebraic geometry code of length n is defined using two divisors G and D = P1+ ⋯ + Pn. We characterize self-orthogonality of the genus zero code in terms of the divisors G, D and the value of a well-chosen derivative polynomial at points (Pi)1≤i≤n. We explore the existence problem of MDS self-dual codes in the odd characteristic case, and we explicitly construct families of new MDS self-dual codes.

The construction of deletion codes for the editing metric is reduced to the construction of codes over the integers for the Manhattan metric by run length coding. The latter codes are constructed by expurgation of lattices' translates. These lattices, in turn, are obtained from Construction A applied to binary codes and Z4-codes. A lower bound on the size of our codes for the Manhattan distance are obtained through generalized theta series of the corresponding lattices. For any fixed number of deletions, provided the number of runs is large enough our method supplies a correction technique. For fixed number of runs and binary sequence length large our lattice construction is shown to be tight up to constants.

Insertion/Deletion codes for the Levenshtein distance are constructed by truncation of lattices for the L1metric. These lattices are obtained from Construction A applied to binary codes and Z4-codes. Finally, Gilbert and Hamming type of bounds are derived.

In this paper we classify all extremal and s-extremal binary self-dual codes of length 38. There are exactly 2744 extremal self-dual codes, two s-extremal codes, and 1730 s-extremal codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self-dual codes of length 36, and from a generalization of this recursive algorithm for the shadow. The classification of -extremal codes permits to achieve the classification of all -extremal codes with .