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

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.