Two MDS Array Codes for Disk Erasures: the Blaum-Bruck-Vardy Code and the BASIC Code

In this post, I am going to review two erasure codes: the Blaum-Bruck-Vardy code and the BASIC code (also here). These are erasure codes, which means, their purpose is to encode a number of data disks into a number of coding disks so that when one or more data/coding disks fail, the failed disk can be reconstructed using the existing data and coding disks.

A strength of these codes is that although the algebra is described on extension fields/rings over GF(2), the encoding/decoding process uses only Boolean addition/rotation operation and no finite field operation. These codes are also MDS (Maximum Distance Separable), which means they have the largest possible (minimum) distance for a fixed message-length and codeword-length.

(Recall that if a code has d data components and c parity components in its generator matrix in standard form, its distance is at most c + 1 by the Singleton bound. Hence the code is MDS if and only if it can tolerate c arbitrary disk failures.)

The BASIC code does the following things in relations to the BBV code:

  1. Adds a virtual parity bit after each disk, giving each disk an even parity
  2. Does polynomial arithmetic modulo 1+x^p instead of h(x) = 1+x+\cdots + x^{p-1} as in the case of BBV code
  3. Shows equivalence to the BBV code by making a nice observation via Chinese Remainder Theorem
  4. Proves MDS property for any number of coding disks when p is “large enough” and has a certain structure

Open Question: What is the least disk size for which these codes are MDS with arbitrary distance?

Continue reading “Two MDS Array Codes for Disk Erasures: the Blaum-Bruck-Vardy Code and the BASIC Code”

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An Intuitive Explanation of the Discrete Fourier Transform

anEvery time I thought “Now I understand Fourier Transform,” I was wrong.

fourier2
Jean-Baptiste Joseph Fourier, 1768 – 1830. (Image source: Wikipedia)

Doing the Fourier Transform of a function is just seeing it from “another” point of view. The “usual” view of a function is in the standard basis \{e_1, \cdots, e_n\}. For example, f can be seen as a vector (in the basis given by the elements in the domain) whose coordinates are the evaluations of f on the¬†elements in the domain. It can also be seen as a polynomial (in the monomial basis) whose coefficients are these evaluations. Let us call this vector u.

The same function can also be seen from the “Fourier basis”, which is just another orthogonal basis, formed by the basis vectors \{v_t\}, t. The tth coordinate in the new basis will be given by inner product between u and the tth basis vector v_t. We call these inner products the Fourier coefficients. The Discrete Fourier Transform matrix (the DFT matrix) “projects” a function from the standard basis to the Fourier basis in the usual sense of projection: taking the inner product along a given direction.

In this post, I am going to use elementary group theoretic notions, polynomials, matrices, and vectors. The ideas in this post will be similar to this Wikipedia article on Discrete Fourier Transform.

Continue reading “An Intuitive Explanation of the Discrete Fourier Transform”