Research On The Some Topics Of Quantum Codes

Some quantum codes are studied in detail, including a family of quantum codes constracted by Calderbank et al. and the quantum quadratic residue codes. Isometries and equivalences of quantum codes also are researched. Finally, we construct some good nonbinary quantum codes by the graph method.Based on the classical binary simplex code S_m and any fixed-point-free element f of Aut(S_m), Calderbank et al. constructed a family of binary quantum error-correcting codes C_m. They proved that,Aut(C_m) has a normal subgroup H which is a semidirect product group of the centralizer Z(f)of f in GL_m(2) with S_m, and the index [Aut(C_m):H] is the number of elements of F_f ={f,1-f,1/f,1-1/f,1/(1-f),f/(1-f)}that are conjugate to f. In this dissertation, a theorem to describe the relationship between the quotient group Aut(C_m)/H and the set F_f is presented. And a way to find the elements of F_f that are conjugate to f is proposed. Then we prove that the quotient group Aut(C_m)/H is isomorphic to S₃, the symmetric group of order 3, and H is a semidirect product group of GL_m/2 (4) with the classical binary simplex code S_m in the linear case. Finally, we generalize a result due to Calderbank et al..Let p= Am+1 be a fixed prime, and o = Z[δ_p] the integer ring of the real quadratic field Q(δ_p), where a_p = ((?)+1) /2．For all primes l, by reducing modulo l on the cyclic o-module C , Rains constructed the quantum quadratic residue codes C_l, each of which produces a [[p,1,d(l)]] code. Rains proved that, in the split linear case, C_l is the direct sum of C_l⁽¹⁾ X and C_l⁽²⁾ (1-X), where C_l⁽¹⁾ and C_l⁽²⁾ are the associated codes of C_l inGF(l)^p, and X is some element of Mat₂(GF(l)). In this dissertation, it is showed that, in the split linear case, C_l⁽¹⁾ and C_l⁽²⁾ are precisely the expurgated quadratic residue codes (?)and (?), respectively, from which one can study the quantum quadratic residue codes via the quadratic residue codes. The quantum quadratic residue codes are also extended in a nice way, most of which allow PSL₂(l), the projective unimodular group, to act.Some of notions of Bogart et al. are generalized. And some results on the isometries and equivalences of quantum codes are presented, one of which generalizes one of results of Bogart et al. By these results, we construct an isometry of quantum codes which is not an equivalent mapping but preserves the symplectic inner product, i.e. the theorem of MacWilliams cannot be generalized to the quantum codes.A technique to construct nonbinary quantum cyclic codes by graph method given by Schlingemann and Werner with a specific example is presented. We also construct the quantum codes [[8,2,4] ]_p, and ones [[n,n-2,4] ]_p for all odd primes p by graph method.