Permutation-invariant qudit codes from polynomials

A permutation-invariant quantum code on $N$ qudits is any subspace stabilized by the matrix representation of the symmetric group $S_N$ as permutation matrices that permute the underlying $N$ subsystems. When each subsystem is a complex Euclidean space of dimension $q \ge 2$, any permutation-invariant code is a subspace of the symmetric subspace of $(\mathbb C^q)^N.$ We give an algebraic construction of new families of of $d$-dimensional permutation-invariant codes on at least $(2t+1)^2(d-1)$ qudits that can also correct $t$ errors for $d \ge 2$. The construction of our codes relies on a real polynomial with multiple roots at the roots of unity, and a sequence of $q-1$ real polynomials that satisfy some combinatorial constraints. When $N > (2t+1)^2(d-1)$, we prove constructively that an uncountable number of such codes exist.Comment: 14 pages. Minor corrections made, to appear in Linear Algebra and its Application

On Codes of Bounded Trellis Complexity

In this paper, we initiate a structure theory of linear codes with bounded trellis complexity. The theory is based on the observation that the family of linear codes over Fq, some permutation of which has trellis state-complexity at most w, is a minor-closed family. It then follows from a deep result of matroid theory that such codes are characterized by finitely many excluded minors. We provide the complete list of excluded minors for w = 1, and give a partial list for w = 2. We also give a polynomial-time algorithm for determining whether or nor a given code has a permutation with state-complexity at most 1

On Maximum Contention-Free Interleavers and Permutation Polynomials over Integer Rings

An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are of particular interest because they admit analytical designs and simple, practical hardware implementation. Contention-free interleavers have been recently shown to be suitable for parallel decoding of turbo codes. In this correspondence, it is shown that permutation polynomials generate maximum contention-free interleavers, i.e., every factor of the interleaver length becomes a possible degree of parallel processing of the decoder. Further, it is shown by computer simulations that turbo codes using these interleavers perform very well for the 3rd Generation Partnership Project (3GPP) standard.Comment: 13 pages, 2 figures, submitted as a correspondence to the IEEE Transactions on Information Theory, revised versio

On Quadratic Inverses for Quadratic Permutation Polynomials over Integer Rings

An interleaver is a critical component for the channel coding performance of turbo codes. Algebraic constructions are of particular interest because they admit analytical designs and simple, practical hardware implementation. Sun and Takeshita have recently shown that the class of quadratic permutation polynomials over integer rings provides excellent performance for turbo codes. In this correspondence, a necessary and sufficient condition is proven for the existence of a quadratic inverse polynomial for a quadratic permutation polynomial over an integer ring. Further, a simple construction is given for the quadratic inverse. All but one of the quadratic interleavers proposed earlier by Sun and Takeshita are found to admit a quadratic inverse, although none were explicitly designed to do so. An explanation is argued for the observation that restriction to a quadratic inverse polynomial does not narrow the pool of good quadratic interleavers for turbo codes.Comment: Submitted as a Correspondence to the IEEE Transactions on Information Theory Submitted : April 1, 2005 Revised : Nov. 15, 200

Constructions of Rank Modulation Codes

Rank modulation is a way of encoding information to correct errors in flash memory devices as well as impulse noise in transmission lines. Modeling rank modulation involves construction of packings of the space of permutations equipped with the Kendall tau distance. We present several general constructions of codes in permutations that cover a broad range of code parameters. In particular, we show a number of ways in which conventional error-correcting codes can be modified to correct errors in the Kendall space. Codes that we construct afford simple encoding and decoding algorithms of essentially the same complexity as required to correct errors in the Hamming metric. For instance, from binary BCH codes we obtain codes correcting $t$ Kendall errors in $n$ memory cells that support the order of $n!/(\log_2n!)^t$ messages, for any constant $t= 1,2,...$ We also construct families of codes that correct a number of errors that grows with $n$ at varying rates, from $\Theta(n)$ to $\Theta(n^{2})$. One of our constructions gives rise to a family of rank modulation codes for which the trade-off between the number of messages and the number of correctable Kendall errors approaches the optimal scaling rate. Finally, we list a number of possibilities for constructing codes of finite length, and give examples of rank modulation codes with specific parameters.Comment: Submitted to IEEE Transactions on Information Theor