Constant dimension codes from Riemann-Roch spaces

Some families of constant dimension codes arising from Riemann-Roch spaces associated to particular divisors of a curve $\X$ are constructed. These families are generalizations of the one constructed by Hansen


Introduction
Let V = F N q be an N-dimensional vector space over F q , q any prime power. The set P(V ) of all subspaces of V forms a metric space with respect to the subspace distance defined by d s (U, U ′ ) = dim(U +U ′ )−dim(U ∩U ′ ); see [10]. In this general setting, a subspace code C is a subset of the set P(V ). Moreover, if all the subspaces of C have a fixed dimension ℓ, then C is called constant dimension code (or Grassmannian code) and C is a subset of G(ℓ, N)(F q ) the set of all the ℓ-dimensional subspaces of V = F N q . Recently, there has been a lot of interest in codes whose codewords are vector subspaces of a given vector space over F q , since they have been proposed for error control in random linear network coding; see [10]. For general results on bounds and constructions of constant-dimension subspaces codes, see [1-6, 8, 9, 11, 12].
In this paper we describe some families of constant dimension codes arising from algebraic curves over finite fields. Namely, the codewords of these codes will be Riemann-Roch spaces associated to particular divisors. The families we will present are a generalization of the one presented by Hansen; see [7].

Hansen's construction
First of all we recall the definition of constant dimension codes and the related parameters.
The size of the code is denoted by |C| and the minimum distance by The linear network code C is said to be of type [N, ℓ, log q |C|, D(C)]. Its normalized weight is λ(C) = ℓ N , its rate is R(C) = log q (|C|) N l and its normalized minimal distance is δ(C) = D(C) 2ℓ . Now we present the construction due to Hansen; see [7]. Let X be an absolutely irreducible, projective algebraic curve of genus g defined over F q and X(F q ) the set of the F q -rational places of X . Also, let n = |X (F q )|. Fix a positive integer k and consider k P ∈X (Fq) = kD, the Frobenius invariant divisor of degree kn having as support the set of all of the F q -rational places of X . The ambient vector space W of this family of linear network codes will be the Riemann-Roch space If nk > 2g − 2 from the Riemann-Roch theorem we have that dim Fq W = kn + 1 − g = N.
Let s be a fixed non-negative integer. The family H of linear network codes presented in [7] is defined as follows.
Definition 2.2. Let V s = P ∈S P | S ⊆ X (F q ), |S| = s . The family H is given by Since each divisor in V s has degree s, by the Riemann-Roch theorem, if ks > 2g − 2 then each codeword of H k,s has dimension ks + 1 − g.
Hansen [7] determined the parameters of the code H k,s . We summarize its results in the following theorem. Also, normalized weight, rate, and normalized minimal distance are

Some generalizations
We generalized the family of linear network codes H k,s , basically by considering sets of divisors of fixed degree s of size larger than |V s | (see Definition 2.2). In this section we present three families, which can be seen as a generalizations of H k,s .

The family A k,s
We consider divisors of fixed degree s having non-negative weights.
The family A k,s is given by Note that in this case the ambient space is larger than in the case of family H k,s , since each codeword of A k,s is contained in W = L (ksD). Also, if nks > 2g − 2, by the Riemann-Roch theorem, dim Fq (L (ksD)) = nks + 1 − g.
Theorem 3.2. Let A k,s be the linear network code as in Definition 3.1.
Then A k,s is a nks + 1 − g, ks Also, normalized weight, rate, and normalized minimal distance are Proof. By our assumptions k > 2g − 2, which implies nks > 2g − 2 and therefore dim Fq W = nks + 1 − g. Also, each codeword of A k,s has dimension over F q equal to ks + 1 − g. The number of codewords is exactly the number of solutions of the linear equation It also corresponds to the number of s-combinations with repetitions of n elements, namely n+s−1 s . In order to compute the minimum distance of this code, first note that for any two divisors V 1 and V 2 in V ′ s , with and therefore This implies that if k > 2g − 2 then If s = 1, then the intersection between two different spaces V 1 and V 2 has dimension 0. From the definition of the metric we have that: Consider now s > 1 and let V 1 and V 2 be two distinct codewords. Therefore there exists a place P ∈ X (F q ) such that m P = m P . This implies that dim( Finally, note that the following two codewords V 1 = L(k(s − 1)P + kQ) and V 2 = L(k(s − 1)P + kR), with P, Q, R pairwise distinct places of X , have distance equal to 2k. This means that the minimum distance of the code A k,s is exactly 2k.
Concerning the normalized weight, rate, and normalized minimal distance, their computations are straightforward. The estimate on δ(A k,s ) follows from the fact that k The assumption k > 2g − 2 is necessary to know the exact dimension of We can observe that in the construction of codes H k,s the divisors in V s correspond to s-subsets of the set of all the F q -rational places of X ; here the divisors in V ′ s are in bijection with the s-multisubsets of X (F q ). This shows the first difference between H k,s and A k,s . In fact, in the first case the parameter s can be at most n, whereas in the second case we can allow s to be greater than n. So, in principle the construction A k,s can be also applied to curves X not having a large number of F q -rational places.

The family B k,s
In this case we consider divisors of fixed degree s having non-negative weights bounded by another constant w. In the case w = s this new family B k,s,s coincides with A k,s . The purpose of this generalization is to bound the dimension of the ambient space. The family B k,s,w is given by In order to compute the number of codewords of the code B k,s,w we will use the following result.
Theorem 3.5 (Wu, [13]). Let n, s, w be non-negative integers satisfying 0 < w ≤ s ≤ nw. The number of solutions of the linear equation where t = min n, s b+1 .
The following theorem describes the parameters of the codes of family B k,s,w . The proof is very similar to the proof of Theorem 3.2 and therefore we omit it. We used Theorem 3.5 in order to compute the number of codewords in B k,s,w .
Theorem 3.6. Let B k,s,w be the linear network code as in Definition 3.4.

The family C k,s,w
Our last generalization takes into account the fact that allowing the divisors of the fixed degree s to have also negative weights increases the number of codewords without changing the dimension of the ambient space. In order to compute the parameters of this new family we need the following corollary to Theorem 3.5.
The number of solution of the diophantine equation is given by where t = min n, ⌊ s−na b−a+1 ⌋ .
Proof. First note that Equation (6) is equivalent to By Theorem 3.5, the number of solutions of this last equation is where t = min n, ⌊ s−na b−a+1 ⌋ . Definition 3.8. Let k, s, w be fixed positive integers, with 0 < w ≤ s ≤ nw. Let The family C k,s,w is given by Remark 3.9. In Definition 3.8 we restrict ourself to the case m P ∈ {s − w(n − 1), . . . , w} since if for some P ∈ X (F q ) such that m P < s−w(n−1), then P ∈X (Fq) m P = P =P ∈X (Fq) m P + m P < (n − 1)w + s − w(n − 1) = s.
Theorem 3.10. Let C k,s,w be the linear network code as in Definition 3.8. Assume k, s, w are positive integers satisfying k > 2g − 2 and 0 < w ≤ s ≤ nw.

Some comparisons
In this section we present some computations on the rates of the three families described in the paper and of the family H k,s (see Definition 2.2). Due to the shape of the formula log q U ′ n,s,s−w(n−1),w in Theorem 3.10, we gave some restrictions on the values of the parameters n, s, w, in order to handle it. In Table 1 we summarize the normalized weight, the rate, and the normalized minimal distance of the four families.