1,262 research outputs found

    Parity of Sets of Mutually Orthogonal Latin Squares

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    Every Latin square has three attributes that can be even or odd, but any two of these attributes determines the third. Hence the parity of a Latin square has an information content of 2 bits. We extend the definition of parity from Latin squares to sets of mutually orthogonal Latin squares (MOLS) and the corresponding orthogonal arrays (OA). Suppose the parity of an OA(k,n)\mathrm{OA}(k,n) has an information content of dim(k,n)\dim(k,n) bits. We show that dim(k,n)(k2)1\dim(k,n) \leq {k \choose 2}-1. For the case corresponding to projective planes we prove a tighter bound, namely dim(n+1,n)(n2)\dim(n+1,n) \leq {n \choose 2} when nn is odd and dim(n+1,n)(n2)1\dim(n+1,n) \leq {n \choose 2}-1 when nn is even. Using the existence of MOLS with subMOLS, we prove that if dim(k,n)=(k2)1\dim(k,n)={k \choose 2}-1 then dim(k,N)=(k2)1\dim(k,N) = {k \choose 2}-1 for all sufficiently large NN. Let the ensemble of an OA\mathrm{OA} be the set of Latin squares derived by interpreting any three columns of the OA as a Latin square. We demonstrate many restrictions on the number of Latin squares of each parity that the ensemble of an OA(k,n)\mathrm{OA}(k,n) can contain. These restrictions depend on nmod4n\mod4 and give some insight as to why it is harder to build projective planes of order n2mod4n \not= 2\mod4 than for n2mod4n \not= 2\mod4. For example, we prove that when n2mod4n \not= 2\mod 4 it is impossible to build an OA(n+1,n)\mathrm{OA}(n+1,n) for which all Latin squares in the ensemble are isotopic (equivalent to each other up to permutation of the rows, columns and symbols)

    Low-Density Parity-Check Codes From Transversal Designs With Improved Stopping Set Distributions

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    This paper examines the construction of low-density parity-check (LDPC) codes from transversal designs based on sets of mutually orthogonal Latin squares (MOLS). By transferring the concept of configurations in combinatorial designs to the level of Latin squares, we thoroughly investigate the occurrence and avoidance of stopping sets for the arising codes. Stopping sets are known to determine the decoding performance over the binary erasure channel and should be avoided for small sizes. Based on large sets of simple-structured MOLS, we derive powerful constraints for the choice of suitable subsets, leading to improved stopping set distributions for the corresponding codes. We focus on LDPC codes with column weight 4, but the results are also applicable for the construction of codes with higher column weights. Finally, we show that a subclass of the presented codes has quasi-cyclic structure which allows low-complexity encoding.Comment: 11 pages; to appear in "IEEE Transactions on Communications

    Absorbing Set Analysis and Design of LDPC Codes from Transversal Designs over the AWGN Channel

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    In this paper we construct low-density parity-check (LDPC) codes from transversal designs with low error-floors over the additive white Gaussian noise (AWGN) channel. The constructed codes are based on transversal designs that arise from sets of mutually orthogonal Latin squares (MOLS) with cyclic structure. For lowering the error-floors, our approach is twofold: First, we give an exhaustive classification of so-called absorbing sets that may occur in the factor graphs of the given codes. These purely combinatorial substructures are known to be the main cause of decoding errors in the error-floor region over the AWGN channel by decoding with the standard sum-product algorithm (SPA). Second, based on this classification, we exploit the specific structure of the presented codes to eliminate the most harmful absorbing sets and derive powerful constraints for the proper choice of code parameters in order to obtain codes with an optimized error-floor performance.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1306.511

    Tree-Based Construction of LDPC Codes Having Good Pseudocodeword Weights

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    We present a tree-based construction of LDPC codes that have minimum pseudocodeword weight equal to or almost equal to the minimum distance, and perform well with iterative decoding. The construction involves enumerating a dd-regular tree for a fixed number of layers and employing a connection algorithm based on permutations or mutually orthogonal Latin squares to close the tree. Methods are presented for degrees d=psd=p^s and d=ps+1d = p^s+1, for pp a prime. One class corresponds to the well-known finite-geometry and finite generalized quadrangle LDPC codes; the other codes presented are new. We also present some bounds on pseudocodeword weight for pp-ary LDPC codes. Treating these codes as pp-ary LDPC codes rather than binary LDPC codes improves their rates, minimum distances, and pseudocodeword weights, thereby giving a new importance to the finite geometry LDPC codes where p>2p > 2.Comment: Submitted to Transactions on Information Theory. Submitted: Oct. 1, 2005; Revised: May 1, 2006, Nov. 25, 200

    Parity of transversals of Latin squares

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    We introduce a notion of parity for transversals, and use it to show that in Latin squares of order 2mod42 \bmod 4, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving E1,,EnE_1,\dots, E_n, where EiE_i is the number of diagonals of a given Latin square that contain exactly ii different symbols. Let A(ij)A(i\mid j) denote the matrix obtained by deleting row ii and column jj from a parent matrix AA. Define tijt_{ij} to be the number of transversals in L(ij)L(i\mid j), for some fixed Latin square LL. We show that tabtcdmod2t_{ab}\equiv t_{cd}\bmod2 for all a,b,c,da,b,c,d and LL. Also, if LL has odd order then the number of transversals of LL equals tabt_{ab} mod 2. We conjecture that tac+tbc+tad+tbd0mod4t_{ac} + t_{bc} + t_{ad} + t_{bd} \equiv 0 \bmod 4 for all a,b,c,da,b,c,d. In the course of our investigations we prove several results that could be of interest in other contexts. For example, we show that the number of perfect matchings in a kk-regular bipartite graph on 2n2n vertices is divisible by 44 when nn is odd and k0mod4k\equiv0\bmod 4. We also show that perA(ac)+perA(bc)+perA(ad)+perA(bd)0mod4{\rm per}\, A(a \mid c)+{\rm per}\, A(b \mid c)+{\rm per}\, A(a \mid d)+{\rm per}\, A(b \mid d) \equiv 0 \bmod 4 for all a,b,c,da,b,c,d, when AA is an integer matrix of odd order with all row and columns sums equal to k2mod4k\equiv2\bmod4

    Structure of the sets of mutually unbiased bases with cyclic symmetry

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    Mutually unbiased bases that can be cyclically generated by a single unitary operator are of special interest, since they can be readily implemented in practice. We show that, for a system of qubits, finding such a generator can be cast as the problem of finding a symmetric matrix over the field F2\mathbb{F}_2 equipped with an irreducible characteristic polynomial of a given Fibonacci index. The entanglement structure of the resulting complete sets is determined by two additive matrices of the same size.Comment: 20 page
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