## Minimum distance of error correcting codes versus encoding complexity, symmetry, and pseudorandomness

##### Author(s)

Bazzi, Louay Mohamad Jamil, 1974-
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##### Other Contributors

Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.

##### Advisor

Sanjoy K. Mitter and Daniel A. Spielman.

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Show full item record##### Abstract

We study the minimum distance of binary error correcting codes from the following perspectives: * The problem of deriving bounds on the minimum distance of a code given constraints on the computational complexity of its encoder. * The minimum distance of linear codes that are symmetric in the sense of being invariant under the action of a group on the bits of the codewords. * The derandomization capabilities of probability measures on the Hamming cube based on binary linear codes with good distance properties, and their variations. Highlights of our results include: * A general theorem that asserts that if the encoder uses linear time and sub-linear memory in the general binary branching program model, then the minimum distance of the code cannot grow linearly with the block length when the rate is nonvanishing. * New upper bounds on the minimum distance of various types of Turbo-like codes. * The first ensemble of asymptotically good Turbo like codes. We prove that depth-three serially concatenated Turbo codes can be asymptotically good. * The first ensemble of asymptotically good codes that are ideals in the group algebra of a group. We argue that, for infinitely many block lengths, a random ideal in the group algebra of the dihedral group is an asymptotically good rate half code with a high probability. * An explicit rate-half code whose codewords are in one-to-one correspondence with special hyperelliptic curves over a finite field of prime order where the number of zeros of a codeword corresponds to the number of rational points. (cont.) * A sharp O(k-1/2) upper bound on the probability that a random binary string generated according to a k-wise independent probability measure has any given weight. * An assertion saying that any sufficiently log-wise independent probability measure looks random to all polynomially small read-once DNF formulas. * An elaborate study of the problem of derandomizability of AC₀ by any sufficiently polylog-wise independent probability measure. * An elaborate study of the problem of approximability of high-degree parity functions on binary linear codes by low-degree polynomials with coefficients in fields of odd characteristics.

##### Description

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2003. Includes bibliographical references (leaves 207-214). This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.

##### Date issued

2003##### Department

Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.##### Publisher

Massachusetts Institute of Technology

##### Keywords

Electrical Engineering and Computer Science.