5,563 research outputs found
The Treewidth of MDS and Reed-Muller Codes
The constraint complexity of a graphical realization of a linear code is the
maximum dimension of the local constraint codes in the realization. The
treewidth of a linear code is the least constraint complexity of any of its
cycle-free graphical realizations. This notion provides a useful
parametrization of the maximum-likelihood decoding complexity for linear codes.
In this paper, we prove the surprising fact that for maximum distance separable
codes and Reed-Muller codes, treewidth equals trelliswidth, which, for a code,
is defined to be the least constraint complexity (or branch complexity) of any
of its trellis realizations. From this, we obtain exact expressions for the
treewidth of these codes, which constitute the only known explicit expressions
for the treewidth of algebraic codes.Comment: This constitutes a major upgrade of previous versions; submitted to
IEEE Transactions on Information Theor
Quantum Error Correction via Less Noisy Qubits
Known quantum error correction schemes are typically able to take advantage of only a limited class of classical error-correcting codes. Entanglement-assisted quantum error correction is a partial solution which made it possible to exploit any classical linear codes over the binary or quaternary finite field. However, the known entanglement-assisted scheme requires noiseless qubits that help correct quantum errors on noisy qubits, which can be too severe an assumption. We prove that a more relaxed and realistic assumption is sufficient by presenting encoding and decoding operations assisted by qubits on which quantum errors of one particular kind may occur. As in entanglement assistance, our scheme can import any binary or quaternary linear codes. If the auxiliary qubits are noiseless, our codes become entanglement-assisted codes, and saturate the quantum Singleton bound when the underlying classical codes are maximum distance separable
Decoding of MDP Convolutional Codes over the Erasure Channel
This paper studies the decoding capabilities of maximum distance profile
(MDP) convolutional codes over the erasure channel and compares them with the
decoding capabilities of MDS block codes over the same channel. The erasure
channel involving large alphabets is an important practical channel model when
studying packet transmissions over a network, e.g, the Internet
Constructions of Generalized Concatenated Codes and Their Trellis-Based Decoding Complexity
In this correspondence, constructions of generalized concatenated (GC) codes with good rates and distances are presented. Some of the proposed GC codes have simpler trellis omplexity than Euclidean geometry (EG), Reed–Muller (RM), or Bose–Chaudhuri–Hocquenghem (BCH) codes of approximately the same rates and minimum distances, and in addition can be decoded with trellis-based multistage decoding up to their minimum distances. Several codes of the same length, dimension, and minimum distance as the best linear codes known are constructed
Generalized weights and bounds for error probability over erasure channels
New upper and lower bounds for the error probability over an erasure channel
are provided, making use of Wei's generalized weights, hierarchy and spectra.
In many situations the upper and lower bounds coincide and this allows
improvement of existing bounds. Results concerning MDS and AMDS codes are
deduced from those bounds
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