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Averaging bounds for lattices and linear codes
 IEEE Trans. Information Theory
, 1997
"... Abstract — General random coding theorems for lattices are derived from the Minkowski–Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski–Hlawka theorem itself is obtained as the limit, for p!1,ofa ..."
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Abstract — General random coding theorems for lattices are derived from the Minkowski–Hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out. A new version of the Minkowski–Hlawka theorem itself is obtained as the limit, for p!1,ofasimple lemma for linear codes over GF (p) used with plevel amplitude modulation. The relation between the combinatorial packing of solid bodies and the informationtheoretic “soft packing ” with arbitrarily small, but positive, overlap is illuminated. The “softpacking” results are new. When specialized to the additive white Gaussian noise channel, they reduce to (a version of) the de Buda–Poltyrev result that spherically shaped lattice codes and adecoder that is unaware of the shaping can achieve the rate 1=2 log2 (P=N).
Cyclic SelfDual Codes
, 1983
"... It is shown that if the automorphism group of a binary selfdual code satisfies a certain condition then the code contains words of weight congruent to 2 modulo 4. In particular, no cyclic binary selfdual code can have all its weights divisible by 4. The number of cyclic binary selfdual codes of l ..."
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Cited by 71 (5 self)
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It is shown that if the automorphism group of a binary selfdual code satisfies a certain condition then the code contains words of weight congruent to 2 modulo 4. In particular, no cyclic binary selfdual code can have all its weights divisible by 4. The number of cyclic binary selfdual codes of length n is determined, and the shortest nontrivial code in this class is shown to have length 14.
Thin lattice coverings
"... Let ^ be a compact body of positive volume in W, starshaped with respect to an interior point, taken to be the origin. For subsets Q of R n, the functional sup lattices A represents the minimum density with which Q. can be covered by a lattice A of translates of < S. We obtain an upper bound on ..."
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Let ^ be a compact body of positive volume in W, starshaped with respect to an interior point, taken to be the origin. For subsets Q of R n, the functional sup lattices A represents the minimum density with which Q. can be covered by a lattice A of translates of < S. We obtain an upper bound on «9L(#, Z n). If the attributes of # are supplemented with convexity, write 3fC instead. We also bound above the classical minimum la nicecovering density of Jtf. No symmetry conditions are imposed on # and Jf. 1.