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On Bounded-Weight Error-Correcting Codes [article]

Russell Bent, Michael Schear, Lane A. Hemaspaandra, Gabriel Istrate
1999 arXiv   pre-print
This paper computationally obtains optimal bounded-weight, binary, error-correcting codes for a variety of distance bounds and dimensions.  ...  We compare the sizes of our codes to the sizes of optimal constant-weight, binary, error-correcting codes, and evaluate the differences.  ...  In a bounded-weight (w) code, every word has at most w ones.  ... 
arXiv:cs/9906001v1 fatcat:ltxncj2bnzalteesynb2uzjvsq

Linear Programming Bounds for Entanglement-Assisted Quantum Error-Correcting Codes by Split Weight Enumerators

Ching-Yi Lai, Alexei Ashikhmin
2018 IEEE Transactions on Information Theory  
Linear programming approaches have been applied to derive upper bounds on the size of classical codes and quantum codes.  ...  On the other hand, we obtain additional constraints on the size of Pauli subgroups for quantum codes, which allow us to improve the linear programming bounds on the minimum distance of small quantum codes  ...  We are thankful to Markus Grassl for comments and suggestions on a previous version of this manuscript.  ... 
doi:10.1109/tit.2017.2711601 fatcat:5yab7y44dbexjegtslenjdrjv4

Upper Bounds on the Number of Errors Corrected by a Convolutional Code

J. Justesen
2004 IEEE Transactions on Information Theory  
We derive upper bounds on the weights of error patterns that can be corrected by a convolutional code with given parameters, or equivalently we give bounds on the code rate for a given set of error patterns  ...  The bounds parallel the Hamming bound for block codes by relating the number of error patterns to the number of distinct syndromes.  ...  We shall derive an upper bound on the rate of a code where the weights of the correctable error patterns are given by T j = b7=4 + j=4c.  ... 
doi:10.1109/tit.2003.822600 fatcat:wy6gvxvhgzbizgd7iot4t7gjt4

On multiple insertion/deletion correcting codes

A.S.J. Helberg, H.C. Ferreira
2002 IEEE Transactions on Information Theory  
The weight spectra and Hamming distance properties of single insertion/deletion error-correcting codes are analyzed.  ...  From these relationships, new bounds are derived and a general construction for multiple insertion/deletion correcting codes is proposed and evaluated.  ...  Bounds on the Insertion/Deletion Error Correcting Capability Using Propositions 5 to 7, it is possible to derive upper and lower bounds on the insertion/deletion error correcting capability of any codebook  ... 
doi:10.1109/18.971760 fatcat:3axt3lo3tjf6pglfennpp7rabu

Undetected error probabilities of binary primitive BCH codes for both error correction and detection

Min-Goo Kim, Jae Hong Lee
1996 IEEE Transactions on Communications  
In this paper, we investigate the undetected error probabilities for bounded-distance decoding of binary primitive BCH codes when they are used for both error correction and detection on a binary symmetric  ...  We obtain bounds on the undetected error probability of binary primitive BCH codes by applying the result to the code and show that bounds are quantified by the deviation factor of true weight distribution  ...  However, if weight distribution of a binary linear code is binomial-like in a range of weight h, then PE(h) in (2) becomes a constant which is dependent on the error correcting capability of the code.  ... 
doi:10.1109/26.494301 fatcat:5eozhs4hqvaordsfessb6mznz4

Trade-off between the tolerance of located and unlocated errors in nondegenerate quantum error-correcting codes [article]

Henry L. Haselgrove, Peter P. Rohde
2007 arXiv   pre-print
We extend the counting argument behind the well-known quantum Hamming bound to derive a bound on the weights of combinations of located and unlocated errors which are correctable by nondegenerate quantum  ...  Numerical results show that the bound gives an excellent prediction to which combinations of unlocated and located errors can be corrected with high probability by certain large degenerate codes.  ...  On the other hand, the quantum Hamming bound places a limit on which error weights are correctable with certainty.  ... 
arXiv:quant-ph/0605183v2 fatcat:eqyrldzxwjbgtor3xswg5uaibq

Linear programming bounds for quantum amplitude damping codes [article]

Yingkai Ouyang, Ching-Yi Lai
2020 arXiv   pre-print
While quantum weight enumerators establish some of the best upper bounds on the minimum distance of quantum error-correcting codes, these bounds do not directly apply to AQEC codes.  ...  Given that approximate quantum error-correcting (AQEC) codes have a potentially better performance than perfect quantum error correction codes, it is pertinent to quantify their performance.  ...  While quantum weight enumerators establish some of the best upper bounds on the minimum distance of quantum error-correcting codes, these bounds do not directly apply to AQEC codes.  ... 
arXiv:2001.03976v1 fatcat:okq4hlzp5jcjplpr4ar2axutxq

Perfect Mannheim, Lipschitz and Hurwitz weight codes [article]

Murat Güzeltepe
2012 arXiv   pre-print
In this paper, upper bounds on codes over Gaussian integers, Lipschitz integers and Hurwitz integers with respect to Mannheim metric, Lipschitz and Hurwitz metric are given.  ...  We first obtain an upper bound on the number of parity check digits for one Hurwitz error correcting codes over H π .  ...  integers with respect to Lipschitz metric 3.1 Perfect codes correcting errors of Lipschitz weight 1 We first obtain an upper bound on the number of parity check digits for one Lipschitz error correcting  ... 
arXiv:1201.3315v1 fatcat:dct6drti2jhtlkn26iwzuoi7xy

Codes Correcting Limited Patterns of Random Errors Using S-K Metric

Bhu Dev Sharma, Ankita Gaur
2013 Cybernetics and Information Technologies  
The paper gives upper bounds on the codeword lengths for various kinds of "random limited error patterns".  ...  Coding is essential in all communications and in all multi-operation devices, and errors do occur. For error control, the method in vogue is to use code words with redundant digits.  ...  Suppose that we want to correct the error in two positions of weight 1, then (by Theorem 1) we have Upper bound on n with a Hamming-single error Table 5 r 5 Upper bound on n with S-K single error  ... 
doi:10.2478/cait-2013-0004 fatcat:ut6nwxvqbrbx7lqqvxtftfim7u

A probabilistic algorithm for computing minimum weights of large error-correcting codes

J.S. Leon
1988 IEEE Transactions on Information Theory  
In order to build a cryptosystem based on error-correcting codes, we need a family of A probabilistic algorithm for computing minimum weights of large. very large) degrees r, which we use to show that  ...  of large error-correcting codes. code C over Fq of length n and dimension k capable of correction w errors.  ...  minimum weights of large error-correcting codes.  ... 
doi:10.1109/18.21270 fatcat:btwyy5yivncpbbs7lszli6zqni

On the Triple-Error-Correcting Cyclic Codes with Zero Set {1, 2 i + 1, 2 j + 1} [chapter]

Vincent Herbert, Sumanta Sarkar
2011 Lecture Notes in Computer Science  
We investigate their weight distribution via their duals and observe that they have the same weight distribution as 3-error-correcting BCH codes for m < 14.  ...  the 3-error-correcting BCH code.  ...  Acknowledgments: The authors are highly grateful to Daniel Augot and Pascale Charpin for their valuable suggestions and comments on this work.  ... 
doi:10.1007/978-3-642-25516-8_6 fatcat:3l5keyuck5guladunvskhwuzv4

Bounds on the undetected error probabilities of linear codes for both error correction and detection

Mao-Chao Lin
1990 IEEE Transactions on Information Theory  
Abstract -The (n, k , d 2 2t + 1) binary linear codes are studied, which are used for correcting error patterns of weight at most t and detecting other error patterns over a binary symmetric channel.  ...  Using this bound, we comment on some known codes.  ...  is used to correct all the error patterns of weight at most An and to detect other error patterns).  ... 
doi:10.1109/18.57213 fatcat:bd4f7jyprfeyjji4jezemu3fsy

Soft and Hard Decision Decoding Performance [chapter]

Martin Tomlinson, Cen Jung Tjhai, Marcel A. Ambroze, Mohammed Ahmed, Mubarak Jibril
2017 Signals and Communication Technology  
Examining the standard array, it can be seen that the code can correct all single error sequences, some two error sequences and one three error sequence.  ...  Some received sequences on rows with coset leaders with weight greater than t are also corrected.  ... 
doi:10.1007/978-3-319-51103-0_2 fatcat:ln5m34zyz5drnlylcvhgexcwye

Upper Bounds on the Number of Errors Corrected by the Koetter–Vardy Algorithm

J. Justesen
2007 IEEE Transactions on Information Theory  
Index Terms-Reed-Solomon (RS) codes, soft-decision decoding.  ...  By introducing a few simplifying assumptions we derive a simple condition for successful decoding using the Koetter-Vardy algorithm for soft-decision decoding of Reed-Solomon codes.  ...  BOUNDS ON RATE AND CORRECTABLE ERRORS The bound (1) refers to the classical channel model where the transition probabilities are given, the rate of the code is bounded, and the reliability is improved  ... 
doi:10.1109/tit.2007.901169 fatcat:sk7qb3t57bcbtn5p5voz24uacm

Uncorrectable errors of weight half the minimum distance for binary linear codes

Kenji Yasunaga, Toru Fujiwara
2008 2008 IEEE International Symposium on Information Theory  
⇒  Derive a lower bound on .  ...  one for all cosets, ⇒ Correctable/uncorrectable errors have a monotone structure. 13 Monotone Error Structure  Notation  Support of v : S(v) = { i : v i ≠ 0 }  v is covered by u : S(v) ⊆ S(u)   ...   A lower bound on #(correctable errors of weight ) for binary linear codes satisfying some condition.  The bound asymptotically coincides with the upper bound for Reed-Muller codes and random linear  ... 
doi:10.1109/isit.2008.4595132 dblp:conf/isit/YasunagaF08 fatcat:67gztrfaore57oekjfwwh3edye
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