Sumsets in Vector Spaces over Finite Fields

Eliahou, Shalom; Kervaire, Michel

doi:10.1006/jnth.1998.2235

Cited by 39 publications

(31 citation statements)

References 6 publications

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“…Generalizations and alternative proofs of Yuzvinsky's Theorem appear in [1], [3], [5]. The theorem is related to our results via a simple formula for the Hopf-Stiefel function recently proved by Plagne [10].…”

Section: Introductionmentioning

confidence: 63%

“…The canonical coloring extends to complete bipartite graphs in a natural way: if A, B ⊆ F k 2 and K(A, B) is the complete bipartite graph with partite sets A and B, then the edgecoloring defined by f (ab) = a + b is a spec. The bound in Yuzvinsky's Theorem is always tight (see [5]); that is, for r, s ≤ 2 k there exist A, B ⊆ F k 2 with |A| = r, |B| = s, and |C| = r • s. Consequently, p(K r,s ) ≤ r • s. We conjecture that equality holds. A direct proof in the graph-theoretic setting would strengthen all cases of Yuzvinsky's Theorem.…”

Section: Introductionmentioning

confidence: 91%

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Optimal strong parity edge-coloring of complete graphs

et al. 2008

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Section: Introductionmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 91%

Optimal strong parity edge-coloring of complete graphs

et al. 2008

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“…Some lower estimates on the cardinality of A+ B in arbitrary Abelian groups were obtained recently by Lev [25,26] and also by Hamidoune, Lladó and Serra [18] in the case A = B. Moreover, some more refined results in elementary Abelian groups have been proved by Eliahou and Kervaire, see [10][11][12].…”

Section: Theorem 4 Ifmentioning

confidence: 97%

An inverse theorem for the restricted set addition in Abelian groups

Károlyi

2005

Journal of Algebra

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Let A be a set of k 5 elements of an Abelian group G in which the order of the smallest nonzero subgroup is larger than 2k − 3. Then the number of different elements of G that can be written in the form a + a , where a, a ∈ A, a = a , is at least 2k − 3, as it has been shown in [Gy. Károlyi, The Erdős-Heilbronn problem in Abelian groups, Israel J. Math. 139 (2004) 349-359]. Here we prove that the bound is attained if and only if the elements of A form an arithmetic progression in G, thus completing the solution of a problem of Erdős and Heilbronn. The proof is based on the so-called 'Combinatorial Nullstellensatz.'  2005 Published by Elsevier Inc.

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“…Thus, finding a closed-form expression for P m is hard, see e.g. the discussion on sumsets in [18], [20], [21]. Because of that, we seek instead efficient bounds and A.…”

Section: Density Evolution Analysismentioning

confidence: 99%

Iterative Decoding of LDPC Codes Over the <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math> </inline-formula>-Ary Partial Erasure Channel

Cohen

Cassuto

2016

IEEE Trans. Inform. Theory

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Abstract-In this paper, we develop a new channel model, which we name the q-ary partial erasure channel (QPEC). The QPEC has a q-ary input, and its output is either the input symbol or a set of M (2 ≤ M ≤ q) symbols, containing the input symbol. This channel serves as a generalization to the binary erasure channel, and mimics situations when a symbol output from the channel is known only partially; that is, the output symbol contains some ambiguity, but is not fully erased. This type of channel is motivated by non-volatile memory multi-level read channels. In such channels the readout is obtained by a sequence of current/voltage measurements, which may terminate with partial knowledge of the stored level. Our investigation is concentrated on the performance of low-density parity-check (LDPC) codes when used over this channel, thanks to their low decoding complexity using belief propagation. We provide the exact QPEC density-evolution equations that govern the decoding process, and suggest a cardinality-based approximation as a proxy. We then provide several bounds and approximations on the proxy density evolutions, and verify their tightness through numerical experiments. Finally, we provide tools for the practical design of LDPC codes for use over the QPEC.

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Sumsets in Vector Spaces over Finite Fields

Cited by 39 publications

References 6 publications

Optimal strong parity edge-coloring of complete graphs

Optimal strong parity edge-coloring of complete graphs

An inverse theorem for the restricted set addition in Abelian groups

Iterative Decoding of LDPC Codes Over the <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math> </inline-formula>-Ary Partial Erasure Channel

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