—We present a general construction of asymmetric quantum codes based on additive codes under the trace Hermi-tian inner product. Various families of additive codes over F4 are used in the construction of many asymmetric quantum codes over F4.
—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… (More)
—The weights in MDS codes of length n and dimension k over the finite field GF (q) are studied. Up to some explicit exceptional cases, the MDS codes with parameters given by the MDS conjecture are shown to contain all k weights in the range n − k + 1 to n. The proof uses the covering radius of the dual code.
—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 asymmetric quantum maximum distance separable (AQMDS) codes. Assuming the validity of the MDS Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for.
We introduce an additive but not F 4-linear map S from F n 4 to F 2n 4 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 θ-cyclic code, a recently… (More)
—Assuming the validity of the MDS Conjecture, the weight distribution of all MDS codes is known. Using a recently-established characterization of asymmetric quantum error-correcting codes, linear MDS codes can be used to construct asymmetric quantum MDS codes with dz ≥ dx ≥ 2 for all possible values of length n for which linear MDS codes over Fq are known… (More)
We solve an open question in code-based cryptography by introducing the first provably secure group signature scheme from code-based assumptions. Specifically, the scheme satisfies the CPA-anonymity and traceability requirements in the random oracle model, assuming the hardness of the McEliece problem, the Learning Parity with Noise problem , and a variant… (More)
—We study a class of Linear Feedback Shift Registers (LFSRs) with characteristic polynomial f (x) = p(x) q(x) where p(x) and q(x) are distinct irreducible polynomials in F2[x]. Important properties of the LFSRs, such as the cycle structure and the adjacency graph, are derived. A method to determine a state belonging to each cycle and a generic algorithm to… (More)
We determine the cycle structure of linear feedback shift register with arbitrary monic characteristic polynomial over any finite field. For each cycle, a method to find a state and a new way to represent the state are proposed.