51,224 research outputs found
QPSK Block-Modulation Codes for Unequal Error Protection
Unequal error protection (UEP) codes find applications in broadcast channels, as well as in other digital communication systems, where messages have different degrees of importance. Binary linear UEP (LUEP) codes combined with a Gray mapped QPSK signal set are used to obtain new efficient QPSK block-modulation codes for unequal error protection. Several examples of QPSK modulation codes that have the same minimum squared Euclidean distance as the best QPSK modulation codes, of the same rate and length, are given. In the new constructions of QPSK block-modulation codes, even-length binary LUEP codes are used. Good even-length binary LUEP codes are obtained when shorter binary linear codes are combined using either the well-known |u¯|u¯+v¯|-construction or the so-called construction X. Both constructions have the advantage of resulting in optimal or near-optimal binary LUEP codes of short to moderate lengths, using very simple linear codes, and may be used as constituent codes in the new constructions. LUEP codes lend themselves quite naturally to multistage decoding up to their minimum distance, using the decoding of component subcodes. A new suboptimal two-stage soft-decision decoding of LUEP codes is presented and its application to QPSK block-modulation codes for UEP illustrated
Stopping Set Distributions of Some Linear Codes
Stopping sets and stopping set distribution of an low-density parity-check
code are used to determine the performance of this code under iterative
decoding over a binary erasure channel (BEC). Let be a binary
linear code with parity-check matrix , where the rows of may be
dependent. A stopping set of with parity-check matrix is a subset
of column indices of such that the restriction of to does not
contain a row of weight one. The stopping set distribution
enumerates the number of stopping sets with size of with parity-check
matrix . Note that stopping sets and stopping set distribution are related
to the parity-check matrix of . Let be the parity-check matrix
of which is formed by all the non-zero codewords of its dual code
. A parity-check matrix is called BEC-optimal if
and has the smallest number of rows. On the
BEC, iterative decoder of with BEC-optimal parity-check matrix is an
optimal decoder with much lower decoding complexity than the exhaustive
decoder. In this paper, we study stopping sets, stopping set distributions and
BEC-optimal parity-check matrices of binary linear codes. Using finite geometry
in combinatorics, we obtain BEC-optimal parity-check matrices and then
determine the stopping set distributions for the Simplex codes, the Hamming
codes, the first order Reed-Muller codes and the extended Hamming codes.Comment: 33 pages, submitted to IEEE Trans. Inform. Theory, Feb. 201
On the binary weight distribution of some Reed-Solomon codes
Consider an (n,k) linear code with symbols from GF(2 sup M). If each code symbol is represented by a m-tuple over GF(2) using certain basis for GF(2 sup M), a binary (nm,km) linear code is obtained. The weight distribution of a binary linear code obtained in this manner is investigated. Weight enumerators for binary linear codes obtained from Reed-Solomon codes over GF(2 sup M) generated by polynomials, (X-alpha), (X-l)(X-alpha), (X-alpha)(X-alpha squared) and (X-l)(X-alpha)(X-alpha squared) and their extended codes are presented, where alpha is a primitive element of GF(2 sup M). Binary codes derived from Reed-Solomon codes are often used for correcting multiple bursts of errors
Hardness of decoding quantum stabilizer codes
In this article we address the computational hardness of optimally decoding a
quantum stabilizer code. Much like classical linear codes, errors are detected
by measuring certain check operators which yield an error syndrome, and the
decoding problem consists of determining the most likely recovery given the
syndrome. The corresponding classical problem is known to be NP-complete, and a
similar decoding problem for quantum codes is also known to be NP-complete.
However, this decoding strategy is not optimal in the quantum setting as it
does not take into account error degeneracy, which causes distinct errors to
have the same effect on the code. Here, we show that optimal decoding of
stabilizer codes is computationally much harder than optimal decoding of
classical linear codes, it is #P
Simple Rate-1/3 Convolutional and Tail-Biting Quantum Error-Correcting Codes
Simple rate-1/3 single-error-correcting unrestricted and CSS-type quantum
convolutional codes are constructed from classical self-orthogonal
\F_4-linear and \F_2-linear convolutional codes, respectively. These
quantum convolutional codes have higher rate than comparable quantum block
codes or previous quantum convolutional codes, and are simple to decode. A
block single-error-correcting [9, 3, 3] tail-biting code is derived from the
unrestricted convolutional code, and similarly a [15, 5, 3] CSS-type block code
from the CSS-type convolutional code.Comment: 5 pages; to appear in Proceedings of 2005 IEEE International
Symposium on Information Theor
The weight distribution and randomness of linear codes
Finding the weight distributions of block codes is a problem of theoretical and practical interest. Yet the weight distributions of most block codes are still unknown except for a few classes of block codes. Here, by using the inclusion and exclusion principle, an explicit formula is derived which enumerates the complete weight distribution of an (n,k,d) linear code using a partially known weight distribution. This expression is analogous to the Pless power-moment identities - a system of equations relating the weight distribution of a linear code to the weight distribution of its dual code. Also, an approximate formula for the weight distribution of most linear (n,k,d) codes is derived. It is shown that for a given linear (n,k,d) code over GF(q), the ratio of the number of codewords of weight u to the number of words of weight u approaches the constant Q = q(-)(n-k) as u becomes large. A relationship between the randomness of a linear block code and the minimum distance of its dual code is given, and it is shown that most linear block codes with rigid algebraic and combinatorial structure also display certain random properties which make them similar to random codes with no structure at all
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