The asymptotic behavior of Grassmannian codes

where n k q is the q-ary Gaussian binomial coefficient. A natural measure of distance in G q (n, k) is the subspace metric [1], [16] given by

for U, V ∈ G q (n, k). We say that C ⊆ G q (n, k) is an (n, M, d, k) q code in the Grassmann space if |C| = M and d S (U, V ) ≥ d for all distinct U, V ∈ C. Such a code C is also called a constant dimension code. The subspaces in C are called codewords. (Note that the distance between any pair of elements of G q (n, k) is even. Because of this, some authors define the distance between subspaces U and V as 1 2 d S (U, V ).) An important observation is the following: a code C in the Grassmann space G q (n, k) has minimum distance 2δ + 2 or more if and only if each subspace in G(n, k -δ) is contained in at most one codeword. There is a ‘dual’ notion to a Grassmannian code, known as a q-covering design: we say that C ⊆ G q (n, k) is a q-covering design C q (n, k, r) if each element of G q (n, r) is contained in at least one element of C. If each element of G q (n, r) is contained in exactly one element of C, we have a Steiner structure, which is both an optimal Grassmannian code and an optimal q-covering design [12], [21]. Codes and designs in the Grassmannian have been studied extensively in the last five years due to the work by Koetter and Kschischang [16] in random network coding, who showed that an (n, M, d, k) q code can correct any t packet insertions and any s packet erasures, as long as 2t + 2s < d. Our goal in this paper is to examine cases in which we can determine the asymptotic behavior of codes and designs in the Grassmannian.

Let A q (n, d, k) denote the maximum number of codewords in an (n, M, d, k) q code. The packing bound is the best known asymptotic upper bound for A q (n, d, k). If we write d = 2δ + 2, we have

This bound is proved by noting that in an (n, M, 2δ + 2, k) q code, each (k -δ)-dimensional subspace can be contained in at most one codeword. Bounds on A q (n, d, k) were given in many papers, e.g. [9], [10], [11], [12], [16], [17], [23], [25], [26], In particular, the well-known Johnson bound for constant weight codes was adapted for constant dimension codes independently in [11], [12], [26] to show that

.

By iterating this bound, using the observation that A q (n, 2δ + 2, k) = 1 for all k ≤ δ, we obtain the iterated Johnson bound:

It is not difficult to see that the iterated Johnson bound is always stronger than the packing bound (indeed, the packing bound may be derived as a simple corollary of the iterated Johnson bound). However, the main goal of this paper is to prove that the packing bound (and so the iterated Johnson bound) is attained asymptotically for fixed k and δ, k ≥ δ, when n tends to infinity. In other words, we will prove the following theorem, in which the term

Theorem 1: Let q, k and δ be fixed integers, with 0 ≤ δ ≤ k and such that q is a prime power. Then

as n → ∞. In fact, the proof of our theorem shows a little more than this: see the proof of the theorem and the comment in the last section of this paper. Our proof of the lower bound is probabilistic, making use of some of the theory of quasirandom hypergraphs. There are known explicit constructions that produce codes whose size is within a constant factor of the packing bound as n → ∞. Currently, the best codes known are the codes of Etzion and Silberstein [9] that are obtained by extending the codes of Silva, Kschischang, and Koetter [22] using a ‘multi-level construction’. If q = 2 and δ = 2, then the ratio between the size of the code and the packing bound is 0.6657, 0.6274, and 0.625 when k = 4, k = 8, and k = 30 respectively, as n tends to infinity. When k = 3, the ratio of 0.7101 in [22] was improved in [10] to 0.7657. The Reed-Solomon-like codes of [16] represented as a lifting of codewords of maximum rank distance codes [22] approach the packing bound as n → ∞ when one of δ or q also tends to infinity [10,Lemma 19]. Theorem 1 shows that there exist codes approaching the packing bound as n → ∞ even when δ and q are fixed; of course, the challenge is now to construct such codes explicitly.

The paper also proves a similar result for q-covering designs. Let C q (n, k, r) denote the minimum number of k-dimensional subspaces in a q-covering design C q (n, k, r). Bounds on C q (n, k, r) can be found in [8], [13]. Setting r = k -δ, the covering bound states that

This bound may be proved by observing that in a C q (n, k, k -δ) covering design each (k -δ)-dimensional subspace must be contained in at least one codeword. The Schönheim bound is an analogous result to the Johnson bound above:

This bound implies the iterated Schönheim bound [13]:

The iterated Schönheim bound is always at least as strong as the covering bound. But the following theorem shows that when k and δ are fixed