Balanced codes

Balanced codes, in which each codeword contains equally many 1's and 0's, are useful in such applications as in optical transmission and optical recording. When balanced codes are used, the same number of 1's and 0's pass through the channel after the transmission of every word, so the channel is in a dc-null state. Optical channels require this property because they employ AC-coupled devices. Line codes, in which codewords may not be balanced, are also used as dc-free codes in such channels. In this thesis we present the research that leads to the following results: 1- Balanced codes These have higher information rate than existing codes yet maintain similar encoding and decoding complexities. 2- Error-correcting balanced codes In many cases, these give higher information rates and more efficient encoding and decoding algorithms than the best-known equivalent codes. 3- DC-Free coset codes A new technique to design dc-free coset codes was developed. These codes have better properties than existing ones. 4- Generalization of balanced codes -- Balanced codes are generalized in three ways among which the first is the most significant: a) Balanced codes with low dc level These codes are designed based on the combined techniques used in (1) and (3) above. A lower dc-level and higher transitions density is achieved at the cost of one extra check bit. These codes are much more attractive, to optical transmission, than the bare-bone balanced codes. b) Non-Binary Balanced Codes Balanced codes over a non-binary alphabet. c) Semi-Balanced Codes -- Codes in which the number of 1's and 0's in every code word differs by at most a certain value. 5- t-EC/AUED coset codes These are t error correcting/all unidirectional error detecting codes. Again the technique in (3) above is used to design t-EC/AUED coset codes. These codes obtain higher information rate than the best-known equivalent codes and yet maintain the same encoding/decoding complexity

Further Progress on the GM-MDS Conjecture for Reed-Solomon Codes

Designing good error correcting codes whose generator matrix has a support constraint, i.e., one for which only certain entries of the generator matrix are allowed to be non-zero, has found many recent applications, including in distributed coding and storage, multiple access networks, and weakly secure data exchange. The dual problem, where the parity check matrix has a support constraint, comes up in the design of locally repairable codes. The central problem here is to design codes with the largest possible minimum distance, subject to the given support constraint on the generator matrix. An upper bound on the minimum distance can be obtained through a set of singleton bounds, which can be alternatively thought of as a cut-set bound. Furthermore, it is well known that, if the field size is large enough, any random generator matrix obeying the support constraint will achieve the maximum minimum distance with high probability. Since random codes are not easy to decode, structured codes with efficient decoders, e.g., Reed-Solomon codes, are much more desirable. The GM-MDS conjecture of Dau et al states that the maximum minimum distance over all codes satisfying the generator matrix support constraint can be obtained by a Reed Solomon code. If true, this would have significant consequences. The conjecture has been proven for several special case: when the dimension of the code k is less than or equal to five, when the number of distinct support sets on the rows of the generator matrix m, say, is less than or equal to three, or when the generator matrix is sparsest and balanced. In this paper, we report on further progress on the GM-MDS conjecture. In particular, we show that the conjecture is true for all m less than equal to six. This generalizes all previous known results (except for the sparsest and balanced case, which is a very special support constraint).Comment: Submitted to ISIT 201

Mutually Uncorrelated Primers for DNA-Based Data Storage

We introduce the notion of weakly mutually uncorrelated (WMU) sequences, motivated by applications in DNA-based data storage systems and for synchronization of communication devices. WMU sequences are characterized by the property that no sufficiently long suffix of one sequence is the prefix of the same or another sequence. WMU sequences used for primer design in DNA-based data storage systems are also required to be at large mutual Hamming distance from each other, have balanced compositions of symbols, and avoid primer-dimer byproducts. We derive bounds on the size of WMU and various constrained WMU codes and present a number of constructions for balanced, error-correcting, primer-dimer free WMU codes using Dyck paths, prefix-synchronized and cyclic codes.Comment: 14 pages, 3 figures, 1 Table. arXiv admin note: text overlap with arXiv:1601.0817

Further Progress on the GM-MDS Conjecture for Reed-Solomon Codes

Designing good error correcting codes whose generator matrix has a support constraint, i.e., one for which only certain entries of the generator matrix are allowed to be nonzero, has found many recent applications, including in distributed coding and storage, multiple access networks, and weakly secure data exchange. The dual problem, where the parity check matrix has a support constraint, comes up in the design of locally repairable codes. The central problem here is to design codes with the largest possible minimum distance, subject to the given support constraint on the generator matrix. An upper bound on the minimum distance can be obtained through a set of singleton bounds, which can be alternatively thought of as a cut-set bound. Furthermore, it is well known that, if the field size is large enough, any random generator matrix obeying the support constraint will achieve the maximum minimum distance with high probability. Since random codes are not easy to decode, structured codes with efficient decoders, e.g., Reed-Solomon codes, are much more desirable. The GM-MDS conjecture of Dau et al states that the maximum minimum distance over all codes satisfying the generator matrix support constraint can be obtained by a Reed Solomon code. If true, this would have significant consequences. The conjecture has been proven for several special case: when the dimension of the code k is less than or equal to five, when the number of distinct support sets on the rows of the generator matrix m, say, is less than or equal to three, or when the generator matrix is sparsest and balanced. In this paper, we report on further progress on the GM-MDS conjecture. 1. In particular, we show that the conjecture is true for all m less than equal to six. This generalizes all previous known results (except for the sparsest and balanced case, which is a very special support constraint)