How hard are Lexicodes ? ∗

L formed by the above procedure is actually a linear code called the lexicode. Let dim(L) = k; then, step (3) is repeated 2 times. So, the dimension of the code is known after the algorithm finishes. However, starting with the all-zero vector 0, and repeating step (3) 2 times (using zero padding to the left of the already constructed codewords whenever necessary) gives us a lexicode of length n. So, we can construct a (k.d) lexicode in the same greedy fashion as an (n, d)-lexicode. We will generally refer to the (n, k, d)-lexicode by Ldk (where the length can be obtained by len(Ldk) = n). By the construction, it is apparent that lexicodes are nested i.e. L1 ⊂ L d 2 ⊂ · · · ⊂ Ldk = L (when vectors are appropriately padded to the left with zeroes to make them of same length). Also, by the nature of construction, lexicodes meet the Gilbert-Varshamov bound i.e. k ≥ n− lg ∑d−1 i=0 ( n i )