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The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(p e , l) (including Z p e). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite… (More)

—We construct new MDS or near-MDS self-dual codes over large finite fields. In particular we show that there exists a Euclidean self-dual MDS code of length n = q over GF (q) whenever q = 2 m (m ≥ 2) using a Reed-Solomon (RS) code and its extension. It turns out that this MDS self-dual code is an extended duadic code. We construct Euclidean self-dual… (More)

LDPC codes are serious contenders to Turbo codes in terms of decoding performance. One of the main problems is to give an explicit construction of such codes whose Tanner graphs have known girth. For a prime power q and m ≥ 2, Lazeb-nik and Ustimenko construct a q-regular bipartite graph D(m, q) on 2q m vertices, which has girth at least 2m/2 + 4. We regard… (More)

Two class association schemes consist of either strongly regular graphs (SRG) or doubly regular tournaments (DRT). We construct self-dual codes from the adjacency matrices of these schemes. This generalizes the construction of Pless ternary Symmetry codes, Karlin binary Double Circulant codes, Calderbank and Sloane quaternary double circulant codes, and… (More)

A binary code with the same weight distribution as its dual code is called formally self-dual (f.s.d.). We only consider f.s.d. even codes (codes with only even weight codewords). We show that any formally self-dual even binary code C of length n not divisible by 8 is balanced. We show that the weight distribution of a balanced near-extremal f.s.d. even… (More)

Linear Complementary Dual codes (LCD) are binary linear codes that meet their dual trivially. We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the family of rings R k. We give a linear programming bound on the largest size of an LCD code of given length and minimum distance. We make a… (More)

LDPC codes are serious contenders to Turbo codes in terms of decoding performance. One of the main problems is to give an explicit construction of such codes whose Tanner graphs have known girth. For a prime power q and m ≥ 2, Lazeb-nik and Ustimenko construct a q-regular bipartite graph D(m, q) on 2q m vertices, which has girth at least 2m/2 + 4. We regard… (More)

We give a new exposition and proof of a generalized CSS construction for nonbinary quantum error-correcting codes. Using this we construct nonbinary quantum stabilizer codes with various lengths, dimensions, and minimum distances from algebraic curves. We also give asymptotically good nonbinary quantum codes from a Garcia–Stichtenoth tower of function… (More)