Over finite fields, Gabidulin codes [1, 2] can be seen as a rank-metric equivalent of Reed–Solomon codes, where instead of evaluating ordinary polynomials, one uses linearized polynomials (i.e., whose only nonzero coefficients are for monomials whose degree is a nonnegative integer power of the field characteristic). To properly generalize this definition to fields of characteristic zero, it was recently suggested in  to employ –polynomials, which are linear combinations of compositions of a generator of the underlying Galois group of the field extension (that must be cyclic).
Independently, there has been a surge of interest lately in constructing sparsest generator matrices for Reed–Solomon and Gabidulin codes [4, 5, 6, 7, 8], for several applications in distributed computing. Since the rows of a generator matrix are codewords, each row cannot contain more than zeros according to the Singleton bound, where is the dimension of the code. The so called GM–MDS conjecture, posed by  and solved by  and , asserts that this maximum number of zeros at every row is attainable, as long as a certain condition regarding the position of zeros is satisfied. Specifically, this condition requires the zero-entries at every set of rows to intersect in at most minus the number of rows in the intersection.
In this paper we complete the picture by showing that the same condition is necessary and sufficient for the existence of sparse generator matrices for Gabidulin codes over fields of characteristic zero. We note that while the proof of the equivalent condition for Reed–Solomon codes is identical for finite fields and fields of characteristic zero, for Gabidulin codes this is not the case, and the proof from  fails over the latter fields. However, by adopting notions from the Reed–Solomon equivalent (the “Simplified GM–MDS conjecture” [7, Thm. 3]), and combining with a variant of the well-known Schwartz–Zippel lemma, we are able to resolve the problem over fields of characteristic zero. Moreover, our proof also provides a randomized construction algorithm whose probability of success can be arbitrarily high; similar randomized construction algorithms exist for the finite variants of the problem, but their probability of success is lower.
Beyond their application in network coding , space-time codes , and cryptography , Gabidulin codes have applications in low rank matrix recovery  (LRMR), which is normally performed over fields of characteristic zero. In this problem, one reconstructs a low-rank matrix from a given set of linear measurements. If these linear measurements are given by multiplication of the unknown matrix by a parity-check matrix of a Gabidulin code, this problem reduces to syndrome decoding of the respective zero codeword. Since the parity-check matrix of a Gabidulin code has a similar structure to that of the generator matrix [3, Prop. 8], our results imply that when performing LRMR with Gabidulin codes, one may employ linear measurements that depend on a small number of entries of the unknown matrix.
The problem is formally stated in Section II, alongside necessary mathematical background. Our main results are summarized in Section III, and proved in Section V by using auxiliary claims given in Section IV.
Let . Denote the dimension of a subspace over a field by and the span of the elements in a set over the field by . The (total) degree of a (multivariate) polynomial is denoted by (e.g. ). For an matrix and , is the submatrix with the rows and columns indexed in and respectively. Let and and when or has a single element, we sometimes write the element only instead of the set.
Ii Problem Setup
In this section we will first provide a brief background on cyclic Galois extensions. Then, we will define rank metric codes and Gabidulin codes. Finally, we will define our problem, namely, finding Gabidulin codes with support constrained generator matrices over a field of characteristic zero.
Ii-a Field extensions
Let be a field extension of finite degree, i.e. the dimension of
as a vector space overis finite, and let . The automorphism group of , , is the set of automorphisms of that fix , i.e.
with the group operation of function composition . If , is called a Galois extension, in which case, is also denoted by and is called the Galois group of .
In this paper, we will focus on cyclic Galois extensions, whose Galois group is a cyclic group of order :
where the automorphism is the generator and for every . Notice that is the identity automorphism.
For example, for finite fields, when and , the Galois group is cyclic of order with the generator automorphism :
For infinite fields, when is the set of rational numbers and , where is the ’th root of unity, is a Galois extension of degree , where is the Euler’s phi function ( is called the ’th cyclotomic field and an interested reader can refer to ). Its Galois group is isomorphic to the multiplicative group of integers modulo . Since is cyclic for , where
is any odd prime andis any positive integer, it follows that for these values of we have that is a cyclic Galois extension of degree . It is also possible to define cyclic extensions of for any degree by considering subfields of for an odd prime such that is divisible by .
Ii-B Rank metric codes
A linear rank metric code, , over a field extension is an –subspace of of dimension with the rank distance
where represent the entries of . By fixing an ordered basis of over , the elements of can be considered as vectors in , and then the codewords (i.e. the elements of ) can be viewed as matrices over . Then, this definition of the rank distance in (1) is equivalent to the minimum of the rank of the matrix representation of a nonzero codeword.
Notice that by definition in (1), the rank distance of can be upper bounded by the Hamming distance, , where is the number of nonzero entries of . Therefore, the Singleton bound can be written for the rank distance as well:
The codes with are called maximum rank distance (MRD), for which we write by omitting . A generator matrix for an code is a matrix over whose rows form a basis for .
Ii-C Gabidulin codes
Gabidulin codes are defined as the row space of the matrix
where and are –linearly independent (notice that this requires ). Note that Gabidulin codes can be seen as evaluation codes of the so-called –polynomials; a –polynomial is a function of the form for , and every codeword in a Gabidulin code is the evaluations of some –polynomial of –degree at most . Note also that the generator matrix can be chosen as the product of any invertible matrix over and the matrix in (3).
Originally, this was defined by Delsarte  and Gabidulin  for the finite fields, when , , and , as the first general constructions of MRD codes over finite fields. Later , it was extended to fields of characteristic zero and it was shown that when is a cyclic Galois extension and is the generator of , this extension of Gabidulin codes also gives an MRD code . In the rest of the paper, we will assume that is a cyclic Galois extension of order and is of characteristic zero.
Ii-D Problem definition
We consider the problem of finding an MRD code whose generator matrix has support constraints. We describe the support constraints through the subsets as
Over finite fields, this problem was studied in  and it was shown that a necessary and sufficient condition for the existence of MRD codes under support constraints described by the is
Over infinite fields, the fact that (5) is necessary can be shown similar to , since MRD codes are also MDS (2), and since the proof in  applies to both finite and infinite fields. However, a similar proof to  cannot be applied to show that (5) is sufficient when has characteristic zero. The reason is that in finite fields, since the generator matrix in (3) consists of entries in the form of polynomials in the ’s, which, in one step of the proof, allows to reduce the problem to a similar one with a smaller parameter, whereas in the characteristic zero, the entries are in the form of –polynomials (defined in ) and applying the same step turns the problem into one of a different kind. Hence, in this paper, we will show that (5) is sufficient for the existence of MRD codes under the support constraints on the generator matrix given in (4) when has characteristic zero.
Iii Main Results
In this section, we present our main results on the existence of MRD codes in characteristic zero (see Theorem 1) and the best achievable rank distance for the cases where there does not exist any (see Corollary 1). Also, we will give a randomized algorithm for the code construction. The proofs of the theorems will be given in Section V.
If the do not satisfy (5), then as given in  and , and hence, an MRD code does not exist. For this case, Corollary 1 below (which is the analog of [4, Thm. 2]) shows that this upper bound is achievable by the subcodes (i.e., the subspaces) of Gabidulin codes.
Define . Then, for any nonempty , we have that . Hence, by Theorem 1, there exists an Gabidulin code with an generator matrix having zeros dictated by . The first rows of will generate a subcode whose rank distance is as good as the Gabidulin code: . Furthermore, is an upper bound on . Therefore, . Hence, . ∎
Iii-a Code Construction
Fix an –basis for and assume that the conditions for the in Theorem 1 are satisfied, i.e. satisfy (5). Then, each has at most elements by applying (5) with . In [5, Thm. 2] and [4, Corollary 3], it is shown that one can keep adding elements to these sets from without violating any of the inequalities in (5) until each has exactly elements. Note that adding elements to these sets will only put more zero constraints on the generator matrix. Therefore, without loss of generality, we can assume that for all along with (5). Then, we construct a generator matrix for a rank metric code in a randomized manner as described below:
[colframe=black,colback=white, sharp corners,colbacktitle=white,coltitle=black,boxrule=0.45pt] Inputs: A finite nonempty set and subsets satisfying (5).
Choose uniformly at random from .
Let for .
Construct as in (3) in terms of .
where is the column vector with at the th entry and ’s elsewhere (Note that ).
Output: The generator matrix .
Let be subsets of size . For a given matrix , a matrix (over the same field as ) satisfying for every and can be given as in (7).
For a fixed , the statement for every is equivalent to the equation . A solution to this equation can be described in terms of the adjugate of the square matrix . Recall that is the transpose of the cofactor matrix and satisfies . Since has an all zero column, we have , which implies . Furthermore, due to the zero column in , the entries of are zero except the first row, whose entries are for ,
Since and , the row vector satisfies . ∎
Furthermore, if are –linearly independent and the matrix is invertible (i.e. , then the code generated by is an Gabidulin code since the row spaces of and are identical. In Theorem 2, we give a lower bound on the probability of this construction giving an MRD code.
Since is infinite, can be arbitrarily large. Therefore, the probability of constructing an MRD code can be arbitrarily close to .
Iv More on Cyclic Galois Extensions
Before moving to the proofs of the theorems, in this section, we will give some useful properties of the automorphisms in .
Iv-a Linear independence of the elements in
Lemma 2 below lists some equivalent conditions to the –linear dependence of the elements of in terms of the automorphisms in . The first two of these conditions can be also seen as a special case of [3, Prop. 5], where the authors give equivalent rank metrics for the elements of , whereas Lemma 2 only claims these rank metrics simultaneously declare rank deficiency (i.e. returns a rank less than ) for a given element of . It is worth noting, as shown by Augot et al. , that the assumption that the extension is cyclic plays an important role in Lemma 2. This is since its proof relies on the fact that fixes only the elements of (i.e. for any , if and only if ), which is the case for the cyclic extensions.
Let , , and
Then, the following are equivalent:
are –linearly dependent.
The columns of are –linearly dependent.
The top minor of is zero, i.e. .
If for some then the claim is trivial, and hence assume that for every .
: Let be the minimum number of columns of that are –linearly dependent and w.l.o.g. assume that
for some unique , which implies that for every . Then, applying to both sides gives , which implies as . Since the ’s are unique it follows that , which implies . Since , we have for .
: If the top minor of is zero, then there exists such that the ’th row of is in the –span of the first rows. By induction, it can be shown that for any , the ’th row is in the span of the first rows. To see how, assume for some , for all . Then, by applying to both sides, it follows that the ’th row is a linear combination of the first rows; hence it is also in the span of the first rows. As a result, , which implies .
: Assume that for some . Then, for any , applying to both sides yields since , which implies . ∎
Iv-B Schwartz–Zippel Lemma for automorphisms
Recall the Schwartz–Zippel Lemma, which states that for a nonzero multivariate polynomial in variables over a field, a point uniformly chosen at random from , where is a nonempty finite subset of this field, will be a root of with probability at most . In this section, we will give an extension of Schwartz–Zippel Lemma for a special type of functions from to . More precisely, for a given multivariate polynomial over in variables (seen as an matrix), we will consider the function and give a bound on the probability of a randomly chosen point being a zero of . Later, this will help us to derive the bound on the probability given in Theorem 2.
Let be an –basis for . Let be a nonzero multivariate polynomial over in variables. Let be defined as in (8) for , where the are independently uniformly chosen at random from a finite nonempty subset . Then,
Define another polynomial as in the variables , , where is an matrix defined as in (8) for . Since is an –basis, the are –linearly independent and by Lemma 2, is invertible. Then, can be also written as . Hence, is also nonzero and . Furthermore, since
where we use since . Now, applying the Schwartz–Zippel Lemma to the polynomial gives . Hence, . ∎
First of all, notice that it is sufficient to prove Theorem 2 since it implies Theorem 1 when is chosen sufficiently large. Assume are chosen as described in Theorem 2. We know that the code with the generator matrix , which satisfies (4) by Lemma 1, is an Gabidulin code if the ’s are –linearly independent and is invertible. Define as in Lemma 2, by which the ’s are –linearly independent iff . Furthermore, since , we have that
Therefore, it is sufficient to show that or that .
In order to show this, we will appeal to Lemma 3. Define the multivariate polynomial
for the variables , seen as an matrix . Then, it suffices to show that . Hence, by Lemma 3, all we need to show is that is a nonzero polynomial with total degree at most .
To show the bound on the degree of , recall the Leibniz formula for the determinant of an square matrix , which is , where is the permutation group of size and is the sign of the permutation . Thus, when the entries of are polynomials, we can write
Hence, since each entry of has degree one. Furthermore, ; hence, . As a result, .
Lemma 4 (Simplified GM–MDS conjecture [7, Thm. 3]111Compared to [7, Thm. 3], in the statement of Lemma 4, the variable is replaced with and the matrix is flipped about its vertical axis, which may only change the sign of the determinant.).
Let be subsets of size . Then, they satisfy (5) if and only if the determinant of the matrix
with entries is not the zero polynomial in the variables .
Notice that the ’th row of in (11) consists of the coefficients of the polynomial
in the variable . We will also show that can be written in the form of (7). To see how, define the Vandermonde matrix . Fix and consider the determinant of the Vandermonde matrix , where is a column vector whose ’th entry is for :
where . On the other hand, by the linearity of the determinant in the first column, we can write
since . As a result, the entries of satisfy
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