Norm Quadratic-Residue Codes

Let p be an odd prime. In the ?nite ?eld GF(p), a distributionproperty of quadratic residues modulus p was given by Perron. On the basisof Perron Theorem, Tiu and Wallace constructed a class of norm quadratic-residue codes Cp of length p(p?1) ,when p is a prime of the form 4m+1. Thisfamily of codes is weakly self-dual, the dimension k of its associated code Cpis at most p, and the minimum distance of it is at least p ? 1. In this thesis,we generalize the norm quadratic-residue codes to arbitrary ?nite ?elds andpartly prove the conjecture of the dimension dimCp = p proposed by Tiu andWallace. Firstly , let p be any odd prime, a new class of binary codes can be con-structed whose generator matrix G is with the following characteristics: 1). Thecoordinate positions are the p(p ? 1) ordered pairs (x,y), where x,y ∈ Fp and √y = 0. 2). The ?rst row R0 of the generator matrix G has a 1 in (x,y) = x+ δyif and only if the Norm x2 ?δy2 is a quadratic residue modp, where δ is a ?xedbut arbitrarily chosen non-residue. 3). For 1 ≤ n ≤ p ? 1, the entries of the(n + 1)th row Rn are obtained by a blockwise cyclic shift in the entries of R0.The (p + 1)th row of G, Rp, is the vector 1 composed of all 1's. Secondly, some fundamental properties of Cp are established. In the gen-erator matrix G of a norm quadratic-residue code Cp, the distribution of 1's inevery block is symmetry, so except for the last row, the weight of every row isp?1 . Accordingly, the minimum distance of a norm quadratic-residue code Cp2 25赋范二次剩余码is at most (p?1)2, and Cp is weakly self-dual. 2 Next, the projective special linear group on GF(p) is applied to codes Cp.PSL2(p) ?xes the norm quadratic-residue code Cp, and acts transitively onthe coordinate position of the code. By these conclusions, we prove that theminimum distance of a norm quadratic-residue code Cp is at least p ? 1. Finally, the conjecture on the dimension of the norm quadratic-residuecodes is partly proved.