In earlier eighties of XX century A.A.Nechaev has obtained the criterion of full period of a Galois polynomial over primary residue ring Z2n . Also he has obtained necessary conditions of maximal period of the Galois polynomial over Z2n in terms of coefficients of this polynomial. Further A.S.Kuzmin has obtained analogous results for the case of Galois polynomial over primary residue ring of od...

We propose two types, namely Type-I and Type-II, quantum stabilizer codes using quadratic residue sets of prime modulus given by the form p = 4n ± 1. The proposed Type-I stabilizer codes are of cyclic structure and code length N = p. They are constructed based on multi-weight circulant matrix generated from idempotent polynomial, which is obtained from a quadratic residue set. The proposed Type...

We show that every Dedekind domain $R$ lying between the polynomial rings $\mathbb Z[X]$ and Q[X]$ with property its residue fields of prime characteristic are finite is equal to a generalized ring integer-valued polynomials, is, for each $p\in\mathbb Z$ there exists subset $E_p$ transcendental elements over Q$ in absolute integral closure $\overline{\mathbb Z_p}$ $p$-adic integers such $R=\{f\...

Recent breakthrough methods [GGMZ, Jou, BGJT] on computing discrete logarithms in small characteristic finite fields share an interesting feature in common with the earlier medium prime function field sieve method [JL]. To solve discrete logarithms in a finite extension of a finite field F, a polynomial h(x) ∈ F[x] of a special form is constructed with an irreducible factor g(x) ∈ F[x] of the d...

This paper studies self-dual and maximal self-orthogonal codes over GF(3). First, a number of Gleason-type theorems are given, describing the weight enumerators of such codes. Second, a table of all such codes of length _-<12 is constructed. Finally, the complete weight enumerators of various quadratic residue and symmetry codes of length =<60.are obtained.

In this study, an efficient and fast algebraic decoding algorithm (ADA) for the binary systematic quadratic residue (QR) code of length 73 with the reducible generator polynomial to correct up to six errors is proposed. The S(I, J ) matrix method given by He et al. (2001) is utilised to compute the unknown syndromes S5. A technique called swap base is proposed to correct the weight-4 error patt...