Quantifying Equivocation for Finite Blocklength Wiretap Codes

This paper presents a new technique for providing the analysis and comparison of wiretap codes in the small blocklength regime over the binary erasure wiretap channel. A major result is the development of Monte Carlo strategies for quantifying a code's equivocation, which mirrors techniques used to analyze normal error correcting codes. For this paper, we limit our analysis to coset-based wiretap codes, and make several comparisons of different code families at small and medium blocklengths. Our results indicate that there are security advantages to using specific codes when using small to medium blocklengths.Comment: Submitted to ICC 201

Cyclic LRC Codes, binary LRC codes, and upper bounds on the distance of cyclic codes

We consider linear cyclic codes with the locality property, or locally recoverable codes (LRC codes). A family of LRC codes that generalize the classical construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Trans. Inform. Theory, no. 8, 2014). In this paper we focus on optimal cyclic codes that arise from this construction. We give a characterization of these codes in terms of their zeros, and observe that there are many equivalent ways of constructing optimal cyclic LRC codes over a given field. We also study subfield subcodes of cyclic LRC codes (BCH-like LRC codes) and establish several results about their locality and minimum distance. The locality parameter of a cyclic code is related to the dual distance of this code, and we phrase our results in terms of upper bounds on the dual distance.Comment: 12pp., submitted for publication. An extended abstract of this submission was posted earlier as arXiv:1502.01414 and was published in Proceedings of the 2015 IEEE International Symposium on Information Theory, Hong Kong, China, June 14-19, 2015, pp. 1262--126

Multiple Particle Interference and Quantum Error Correction

The concept of multiple particle interference is discussed, using insights provided by the classical theory of error correcting codes. This leads to a discussion of error correction in a quantum communication channel or a quantum computer. Methods of error correction in the quantum regime are presented, and their limitations assessed. A quantum channel can recover from arbitrary decoherence of x qubits if K bits of quantum information are encoded using n quantum bits, where K/n can be greater than 1-2 H(2x/n), but must be less than 1 - 2 H(x/n). This implies exponential reduction of decoherence with only a polynomial increase in the computing resources required. Therefore quantum computation can be made free of errors in the presence of physically realistic levels of decoherence. The methods also allow isolation of quantum communication from noise and evesdropping (quantum privacy amplification).Comment: Submitted to Proc. Roy. Soc. Lond. A. in November 1995, accepted May 1996. 39 pages, 6 figures. This is now the final version. The changes are some added references, changed final figure, and a more precise use of the word `decoherence'. I would like to propose the word `defection' for a general unknown error of a single qubit (rotation and/or entanglement). It is useful because it captures the nature of the error process, and has a verb form `to defect'. Random unitary changes (rotations) of a qubit are caused by defects in the quantum computer; to entangle randomly with the environment is to form a treacherous alliance with an enemy of successful quantu

Quasi-cyclic subcodes of cyclic codes

We completely characterize possible indices of quasi-cyclic subcodes in a cyclic code for a very broad class of cyclic codes. We present enumeration results for quasi-cyclic subcodes of a fixed index and show that the problem of enumeration is equivalent to enumeration of certain vector subspaces in finite fields. In particular, we present enumeration results for quasi-cyclic subcodes of the simplex code and duals of certain BCH codes. Our results are based on the trace representation of cyclic codes

Cyclic LRC Codes and their Subfield Subcodes

We consider linear cyclic codes with the locality property, or locally recoverable codes (LRC codes). A family of LRC codes that generalizes the classical construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Transactions on Information Theory, no. 8, 2014; arXiv:1311.3284). In this paper we focus on the optimal cyclic codes that arise from the general construction. We give a characterization of these codes in terms of their zeros, and observe that there are many equivalent ways of constructing optimal cyclic LRC codes over a given field. We also study subfield subcodes of cyclic LRC codes (BCH-like LRC codes) and establish several results about their locality and minimum distance.Comment: Submitted for publicatio