39,978 research outputs found
Multilevel Polarization of Polar Codes Over Arbitrary Discrete Memoryless Channels
It is shown that polar codes achieve the symmetric capacity of discrete
memoryless channels with arbitrary input alphabet sizes. It is shown that in
general, channel polarization happens in several, rather than only two levels
so that the synthesized channels are either useless, perfect or "partially
perfect". Any subset of the channel input alphabet which is closed under
addition, induces a coset partition of the alphabet through its shifts. For any
such partition of the input alphabet, there exists a corresponding partially
perfect channel whose outputs uniquely determine the coset to which the channel
input belongs. By a slight modification of the encoding and decoding rules, it
is shown that perfect transmission of certain information symbols over
partially perfect channels is possible. Our result is general regarding both
the cardinality and the algebraic structure of the channel input alphabet; i.e
we show that for any channel input alphabet size and any Abelian group
structure on the alphabet, polar codes are optimal. It is also shown through an
example that polar codes when considered as group/coset codes, do not achieve
the capacity achievable using coset codes over arbitrary channels
Improved Constructions of Frameproof Codes
Frameproof codes are used to preserve the security in the context of
coalition when fingerprinting digital data. Let be the largest
cardinality of a -ary -frameproof code of length and
. It has
been determined by Blackburn that when ,
when and is even, and . In this paper, we
give a recursive construction for -frameproof codes of length with
respect to the alphabet size . As applications of this construction, we
establish the existence results for -ary -frameproof codes of length
and size for all odd when and for all
when . Furthermore, we show that
meeting the upper bound given by Blackburn, for all integers such that
is a prime power.Comment: 6 pages, to appear in Information Theory, IEEE Transactions o
An improved bound on the zero-error list-decoding capacity of the 4/3 channel
We prove a new, improved upper bound on the size of codes C ⊆{1, 2, 3, 4}n with the property that every four distinct codewords in C have a coordinate where they all differ. Specifically, we show that such a code has size at most 26n/19 +o(n), or equivalently has rate bounded by 6/19 ≤ 0.3158 (measured in bits). This improves the previous best upper bound of 0.3512 due to (Arikan 1994), which in turn improved the 0.375 bound that followed from general bounds for perfect hashing due to (Fredman and Komlos, 1984) and (Korner and Marton, 1988). The context for this problem is two-fold: zero-error list decoding capacity, where such codes give a way to communicate with no error on the “4/3 channel” when list-of-3 decoding is employed, and perfect hashing, where such codes give a perfect hash family of size n mapping C to {1, 2, 3, 4}
Codes for Asymmetric Limited-Magnitude Errors With Application to Multilevel Flash Memories
Several physical effects that limit the reliability and performance of multilevel flash memories induce errors that have low magnitudes and are dominantly asymmetric. This paper studies block codes for asymmetric limited-magnitude errors over q-ary channels. We propose code constructions and bounds for such channels when the number of errors is bounded by t and the error magnitudes are bounded by ℓ. The constructions utilize known codes for symmetric errors, over small alphabets, to protect large-alphabet symbols from asymmetric limited-magnitude errors. The encoding and decoding of these codes are performed over the small alphabet whose size depends only on the maximum error magnitude and is independent of the alphabet size of the outer code. Moreover, the size of the codes is shown to exceed the sizes of known codes (for related error models), and asymptotic rate-optimality results are proved. Extensions of the construction are proposed to accommodate variations on the error model and to include systematic codes as a benefit to practical implementation
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