Bounds on Covering Codes with the Rank Metric

There is a steady stream of works on rank metric codes due to their applications to data storage, publickey cryptosystems, and space-time coding (see [1], [2] and the references therein for a comprehensive literature survey), and interest in rank metric codes is strengthened by their recent applications in error control for random network coding [2], decribed below. Constant-dimension codes [3] are an important class of codes for error and erasure correction in random linear network coding. Using the lifting operation [2], rank metric codes can be readily turned into constant-dimension codes without modifying their distance properties. It is shown in [3] that liftings of rank metric codes have many advantages compared to general constant-dimension codes. First, their cardinalities are optimal up to a constant; second, the decoding of these codes can be done in either the subspace metric [3] or the rank metric [2], and for both scenarios efficient decoding algorithms were proposed.

Despite their significance, many research problems in rank metric codes remain open. For example, geometrical properties of the rank metric space are not well studied, and covering properties of rank metric codes have received little attention with the exception of our previous work [1]. The geometrical properties, such as the intersection number, characterize fundamental properties of a metric space, and determine the packing and covering properties of codes defined in the metric space. Packing and covering properties are significant to the code design, decoding, and performance analysis of rank metric codes.

For instance, the covering radius can be viewed as a measure of performance: if the code is used for error correction, then the covering radius is the maximum weight of a correctable error vector. The covering radius of a code also gives a straightforward criterion for code optimality: if the covering radius of a code is at least equal to its minimum distance, then more codewords can be added without altering the minimum distance, and hence the original code is not optimal. Although liftings of rank metric codes are nearly optimal constant-dimension codes, they have the largest possible covering radius [4], and hence they are not optimal constant-dimension codes. This covering property hence is a crucial result for the design of error control codes for random linear network coding.

In this paper, we focus on the geometrical properties of the rank metric space and covering properties of rank metric codes. We first establish an analytical expression for the intersection of two balls with rank radii, and then derive an upper bound on the volume of the union of multiple balls with rank radii. Both results are novel to the best of our knowledge. Using these geometrical properties, we derive novel lower and upper bounds on the minimum cardinality K R (q m , n, ρ) of a code in GF(q m ) n with rank covering radius ρ. The bounds presented in this paper are obtained based on different approaches from those in [1], and they are the tightest bounds to the best of our knowledge for many sets of parameter values.

The rank weight of a vector x ∈ GF(q m ) n , denoted as rk(x), is defined to be the maximum number of coordinates in x that are linearly independent over GF(q). The number of vectors of rank weight r in

for r ≥ 1 [5]. The rank distance between x and y is defined as d R (x, y) = rk(xy). If a vector x is at distance at most ρ from a code C, we say C covers x with radius ρ. The rank covering radius ρ of a November 4, 2018 DRAFT code C is defined as max x∈GF(q m ) n d R (x, C). When n ≤ m, the minimum rank distance d R of a code of length n and cardinality M over GF(q m ) satisfies d R ≤ nlog q m M + 1; we refer to this bound as the Singleton bound for rank metric codes. The equality is attained by a class of linear rank metric codes called maximum rank distance (MRD) codes.

We denote the intersection of two spheres (balls, respectively) in GF(q m ) n with rank radii u and s and centers with distance w as J R (u, s, w) (I(u, s, w), respectively).

Lemma 1: We have

where K j (i) is a q-Krawtchouk polynomial [6]:

Although ( 1) is obtained by a direct application of [7, Chapter II, Theorem 3.6], we present it formally since it is a fundamental geometric property of the rank metric space. Since

(1) also leads to an analytical expression for I(u, s, w). In order to simplify notations we denote I(ρ, ρ, d)

as I(ρ, d). Both (1) and ( 3) are instrumental in our later derivations.

We denote the volume of a ball with rank radius ρ as v(ρ), and we now derive a bound on the volume of the union of any K balls with radius ρ, which will be instrumental in Section IV.

Lemma 2: The volume of the union of any K balls with rank radius ρ is at most

where l = ⌊log q m K⌋.

Proof: Let {v i } K-1 i=0 denote the centers of K balls with rank radius ρ and let V j = {v i } j-1 i=0 for 1 ≤ j ≤ K. The centers are labeled such that d R (v j , V j ) is

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