Somphong Jitman

CSS-Like Constructions of Asymmetric Quantum Codes

Asymmetric quantum error-correcting codes (AQCs) may offer some advantage over their symmetric counterparts by providing better error-correction for the more frequent error types. The well-known CSS construction of q-ary AQCs is extended by removing the Fq-linearity requirement as well as the limitation on the type of inner product used. The proposed constructions are called CSS-like constructions and utilize pairs of nested subfield linear codes under one of the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products. After establishing some theoretical foundations, best-performing CSS-like AQCs are constructed. Combining some constructions of nested pairs of classical codes and linear programming, many optimal and good pure q-ary CSS-like codes for q ∈ {2,3,4,5,7,8,9} up to reasonable lengths are found. In many instances, removing the Fq-linearity and using alternative inner products give us pure AQCs with improved parameters than relying solely on the standard CSS construction.

PURE ASYMMETRIC QUANTUM MDS CODES FROM CSS CONSTRUCTION: A COMPLETE CHARACTERIZATION

Using the Calderbank–Shor–Steane (CSS) construction, pure q-ary asymmetric quantum error-correcting codes attaining the quantum Singleton bound are constructed. Such codes are called pure CSS asymmetric quantum maximum distance separable (AQMDS) codes. Assuming the validity of the classical maximum distance separable (MDS) Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for.

Hermitian Self-Dual Abelian Codes

Hermitian self-dual abelian codes in a group ring F_q²[G], where F_q² is a finite field of order q² and G is a finite abelian group, are studied. Using the well-known discrete Fourier transform decomposition for a semisimple group ring, a characterization of Hermitian self-dual abelian codes in F_q²[G] is given, together with an alternative proof of necessary and sufficient conditions for the existence of such a code in F_q²[G], i.e., there exists a Hermitian self-dual abelian code in F_q²[G] if and only if the order of G is even and q = 2^l for some positive integer l. Later on, the study is further restricted to the case where F₂^2l [G] is a principal ideal group ring, or equivalently, G ≅ A⊕Z₂k with 2 ≠ |A|. Based on the characterization obtained, the number of Hermitian self-dual abelian codes in F₂2l [A⊕Z₂k] can be determined easily. When A is cyclic, this result answers an open problem of Jia et al. concerning Hermitian self-dual cyclic codes. In many cases, F₂2l [A⊕Z₂k] contains a unique Hermitian self-dual abelian code. The criteria for such cases are determined in terms of l and the order of A. Finally, the distribution of finite abelian groups A such that a unique Hermitian self-dual abelian code exists in F₂2l [A ⊕ Z₂] is established, together with the distribution of odd integers m such that a unique Hermitian self-dual cyclic code of length 2 m over F₂2l exists.

Abelian Codes in Principal Ideal Group Algebras

We study abelian codes in principal ideal group algebras (PIGAs). We first give an algebraic characterization of abelian codes in any group algebra and provide some general results. For abelian codes in a PIGA, which can be viewed as cyclic codes over a semisimple group algebra, it is shown that every abelian code in a PIGA admits generator and check elements. These are analogous to the generator and parity-check polynomials of cyclic codes. A characterization and an enumeration of Euclidean self-dual and Euclidean self-orthogonal abelian codes in a PIGA are given, which generalize recent analogous results for self-dual cyclic codes. In addition, the structures of reversible and complementary dual abelian codes in a PIGA are established, again extending results on reversible and complementary dual cyclic codes. Finally, asymptotic properties of abelian codes in a PIGA are studied. An upper bound for the minimum distance of abelian codes in a non-semisimple PIGA is given in terms of the minimum distance of abelian codes in semisimple group algebras. Abelian codes in a non-semisimple PIGA are then shown to be asymptotically bad, similar to the case of repeated-root cyclic codes.

Constructions of good entanglement-assisted quantum error correcting codes

Entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and fundamental class of codes. They allow for the construction of quantum codes from classical codes by relaxing the duality condition and using pre-shared entanglement between the sender and receiver. However, in general it is not easy to determine the number of shared pairs required to construct an EAQECC. In this paper, we show that this number is related to the hull of the classical code. Using this fact, we give methods to construct EAQECCs requiring desirable amounts of entanglement. This allows for designing families of EAQECCs with good error performance. Moreover, we construct maximal entanglement EAQECCs from LCD codes. Finally, we prove the existence of asymptotically good EAQECCs in the odd characteristic case.

Xing---Ling codes, duals of their subcodes, and good asymmetric quantum codes

A class of powerful

The average dimension of the Hermitian hull of cyclic codes over finite fields of square order

q

Enumeration of Self-Dual Cyclic Codes of some Specific Lengths over Finite Fields

q-ary linear polynomial codes originally proposed by Xing and Ling is deployed to construct good asymmetric quantum codes via the standard CSS construction. Our quantum codes are