Christian Senger

Multitrial decoding of concatenated codes using fixed thresholds

For decoding concatenated codes up to half their designed distance, generalized minimum distance (GMD) decoding can be used. GMD decoding applies multitrial error/erasure decoding of the outer code, where erased symbols depend on some reliability measure stemming from the inner decoders. We consider the case where the outer decoder is able to decode beyond half the minimum distance of the outer code. For a given number of outer decoding trials, we derive achievable decoding radii for GMD decoding. Vice versa, we give a lower bound on the number of required outer decoding trials to obtain the greatest possible decoding radius.

On Extended Forney-Kovalev GMD decoding

Consider a code C with Hamming distance d. Assume we have a decoder Φ that corrects ε errors and θ erasures if λε + θ ≤ d − 1, where a real number 1 ≪ λ ≤ 2 is the tradeoff rate between errors and erasures for this decoder. This holds e.g. for l-punctured Reed-Solomon codes, i.e., codes over the field F <inf>q</inf>^l of length n ≪ q with locators taken from the subfield F<inf>q</inf>, where l ∈ {1, 2, . . .} and λ = 1+1/l. We propose an m-trial generalized minimum distance (GMD) decoder based on λ. Our approach extends results of Forney and Kovalev (obtained for λ = 2) to the whole given range of λ. We consider both fixed erasing and adaptive erasing GMD strategies. For l ≪ 1 the following approximations hold. For the fixed erasing strategy the error correcting radius is ρF ≈ d/2 (1 - l^−m/2). For the adaptive erasing strategy, ρA ≈ d/2 (1 - l^−2m) quickly approaches d/2 if l or m grows. The minimum number of decoding trials required to reach an error correcting radius d/2 is m<inf>A</inf> = 1/2 (log<inf>l</inf> d + 1). This means that 2 or 3 trials are sufficient to reach ρA = d/2 in many practical cases if l ≫ 1.

End-to-End algebraic network coding for wireless TCP/IP networks

The Transmission Control Protocol (TCP) was designed to provide reliable transport services in wired networks. In such networks, packet losses mainly occur due to congestion. Hence, TCP was designed to apply congestion avoidance techniques to cope with packet losses. Nowadays, TCP is also utilized in wireless networks where, besides congestion, numerous other reasons for packet losses exist. This results in reduced throughput and increased transmission round-trip time when the state of the wireless channel is bad. We propose a new network layer, that transparently sits below the transport layer and hides non congestion-imposed packet losses from TCP. The network coding in this new layer is based on the well-known class of Maximum Distance Separable (MDS) codes.

Decoding generalized concatenated codes using Interleaved Reed-Solomon codes

Generalized Concatenated codes are a code construction consisting of a number of outer codes whose code symbols are protected by an inner code. As outer codes, we assume the most frequently used Reed-Solomon codes; as inner code, we assume some linear block code which can be decoded up to half its minimum distance. Decoding up to half the minimum distance of Generalized Concatenated codes is classically achieved by the Blokh-Zyablov-Dumer algorithm, which iteratively decodes by first using the inner decoder to get an estimate of the outer code words and then using an outer error/erasure decoder with a varying number of erasures determined by a set of pre- calculated thresholds. In this paper, a modified version of the Blokh-Zyablov-Dumer algorithm is proposed, which exploits the fact that a number of outer Reed-Solomon codes with average minimum distance d macr can be grouped into one single Interleaved Reed-Solomon code which can be decoded beyond d macr/2. This allows to skip a number of decoding iterations on the one hand and to reduce the complexity of each decoding iteration significantly - while maintaining the decoding performance - on the other.

Single-trial decoding of concatenated codes using fixed or adaptive erasing

We consider a concatenated code with designed distance dodi

The periodicity transform in algebraic decoding of Reed-Solomon codes

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Optimal thresholds for GMD decoding with ℓ+1 over ℓ-extended Bounded Distance decoders

, based on an outer code with distance do and an inner code with distance di. To decode the inner code, we use a Bounded Minimum Distance decoder correcting up to (di

Optimal threshold-based multi-trial error/erasure decoding with the Guruswami-Sudan algorithm

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Asymptotic single-trial strategies for GMD decoding with arbitrary error-erasure tradeoff

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Adaptive single-trial error/erasure decoding of binary codes

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