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By extending the notion of minimum rank distance, this paper introduces two new relative code parameters of a linear code C<sub>1</sub> of length n over a field extension F<sub>q</sub><sup>m</sup> and its subcode C<sub>2</sub> ⊆ C<sub>1</sub>. One is called the relative dimension/intersection profile (RDIP), and the other is called the relative… (More)
We generalize the random coding argument of stabilizer codes and derive a lower bound on the quantum capacity of an arbitrary discrete memoryless quantum channel. For the depolarizing channel, our lower bound coincides with that obtained by Bennett et al. We also slightly improve the quantum Gilbert–Varshamov bound for general stabilizer codes, and… (More)
The secure multiplex coding (SMC) is a technique to remove rate loss in the coding for wiretap channels and broadcast channels with confidential messages caused by the inclusion of random bits into transmitted signals. SMC replaces the random bits by other meaningful secret messages, and a collection of secret messages serves as the random bits to hide the… (More)
The stabilizer code is the most general algebraic construction of quantum error-correcting codes proposed so far. A stabilizer code can be constructed from a self-orthogonal subspace of a symplectic space over a finite field. We propose a construction method of such a self-orthogonal space using an algebraic curve. By using the proposed method we construct… (More)
SUMMARY This paper precisely characterizes secret sharing schemes based on arbitrary linear codes by using the relative dimension/length profile (RDLP) and the relative generalized Hamming weight (RGHW). We first describe the equivocation Δ m of the secret vector s = [s 1 ,...,s l ] given m shares in terms of the RDLP of linear codes. We also characterize… (More)
SUMMARY We generalize the construction of quantum error-correcting codes from F 4-linear codes by Calderbank et al. to p m-state systems. Then we show how to determine the error from a syndrome. Finally we discuss a systematic construction of quantum codes with efficient decoding algorithms.
We propose a method for computing the radical of an arbitrary ideal in the polynomial ring in n variables over a perfect field of characteristic p > 0. In our method Buchberger's algorithm is performed once in n variables and a Gröbner basis conversion algorithm is performed at most n log p d times in 2n variables, where d is the maximum of total degrees of… (More)
SUMMARY We consider the problem of secret key agreement in Gaussian Maurer's Model. In Gaussian Maurer's model, legitimate receivers, Alice and Bob, and a wire-tapper, Eve, receive signals randomly generated by a satellite through three independent memoryless Gaussian channels respectively. Then Al-ice and Bob generate a common secret key from their… (More)
We generalize Sudan's list decoding algorithm without mul-tiplicity to evaluation codes coming from arbitrary order domains. The number of correctable errors by the proposed method is larger than the original list decoding without multiplicity.
We present a new bound on the number of F q-rational places in an algebraic function field. It uses information about the generators of the Weierstrass semigroup related to a rational place. As we demonstrate, the bound has implications to the theory of towers of function fields.