Additive Asymmetric Quantum Codes

We present a general construction of asymmetric quantum codes based on additive codes under the trace Hermitian inner product. Various families of additive codes over \F_{4} are used in the construction of many asymmetric quantum codes over \F_{4}.Comment: Accepted for publication March 2, 2011, IEEE Transactions on Information Theory, to appea

From Skew-Cyclic Codes to Asymmetric Quantum Codes

We introduce an additive but not $\mathbb{F}_4$-linear map $S$ from $\mathbb{F}_4^{n}$ to $\mathbb{F}_4^{2n}$ and exhibit some of its interesting structural properties. If $C$ is a linear $[n,k,d]_4$-code, then $S(C)$ is an additive $(2n,2^{2k},2d)_4$-code. If $C$ is an additive cyclic code then $S(C)$ is an additive quasi-cyclic code of index $2$. Moreover, if $C$ is a module $\theta$-cyclic code, a recently introduced type of code which will be explained below, then $S(C)$ is equivalent to an additive cyclic code if $n$ is odd and to an additive quasi-cyclic code of index $2$ if $n$ is even. Given any $(n,M,d)_4$-code $C$, the code $S(C)$ is self-orthogonal under the trace Hermitian inner product. Since the mapping $S$ preserves nestedness, it can be used as a tool in constructing additive asymmetric quantum codes.Comment: 16 pages, 3 tables, submitted to Advances in Mathematics of Communication

Constructions of Pure Asymmetric Quantum Alternant Codes Based on Subclasses of Alternant Codes

In this paper, we construct asymmetric quantum error-correcting codes(AQCs) based on subclasses of Alternant codes. Firstly, We propose a new subclass of Alternant codes which can attain the classical Gilbert-Varshamov bound to construct AQCs. It is shown that when $d_x=2$, $Z$-parts of the AQCs can attain the classical Gilbert-Varshamov bound. Then we construct AQCs based on a famous subclass of Alternant codes called Goppa codes. As an illustrative example, we get three $[[55,6,19/4]],[[55,10,19/3]],[[55,15,19/2]]$ AQCs from the well known $[55,16,19]$ binary Goppa code. At last, we get asymptotically good binary expansions of asymmetric quantum GRS codes, which are quantum generalizations of Retter's classical results. All the AQCs constructed in this paper are pure

Asymmetric Quantum Codes: New Codes from Old

In this paper we extend to asymmetric quantum error-correcting codes (AQECC) the construction methods, namely: puncturing, extending, expanding, direct sum and the (u|u + v) construction. By applying these methods, several families of asymmetric quantum codes can be constructed. Consequently, as an example of application of quantum code expansion developed here, new families of asymmetric quantum codes derived from generalized Reed-Muller (GRM) codes, quadratic residue (QR), Bose-Chaudhuri-Hocquenghem (BCH), character codes and affine-invariant codes are constructed.Comment: Accepted for publication Quantum Information Processin

Asymmetric quantum codes on non-orientable surfaces

In this paper, we construct new families of asymmetric quantum surface codes (AQSCs) over non-orientable surfaces of genus $g\geq 2$ by applying tools of hyperbolic geometry. More precisely, we prove that if the genus $g$ of a non-orientable surface is even $(g=2h)$, then the parameters of the corresponding AQSC are equal to the parameters of a surface code obtained from an orientable surface of genus $h$. Additionally, if $S$ is a non-orientable surface of genus $g$, we show that the new surface code constructed on a $\{p, q\}$ tessellation over $S$ has the ratio $k/n$ better than the ratio of an AQSC constructed on the same $\{p, q\}$ tessellation over an orientable surface of the same genus $g$

Algebraic structure of F_q-linear conjucyclic codes over finite field F_{q^2}

Recently, Abualrub et al. illustrated the algebraic structure of additive conjucyclic codes over F_4 (Finite Fields Appl. 65 (2020) 101678). In this paper, our main objective is to generalize their theory. Via an isomorphic map, we give a canonical bijective correspondence between F_q-linear additive conjucyclic codes of length n over F_{q^2} and q-ary linear cyclic codes of length 2n. By defining the alternating inner product, our proposed isomorphic map preserving the orthogonality can also be proved. From the factorization of the polynomial x^{2n}-1 over F_q, the enumeration of F_{q}-linear additive conjucyclic codes of length n over F_{q^2} will be obtained. Moreover, we provide the generator and parity-check matrices of these q^2-ary additive conjucyclic codes of length n