1 Introduction and preliminaries
Let denote a finite field, , and let the symmetric group of order . Recall that the determinant of is given by
(1) 
where the sign of the permutation , denoted by , is (resp. ) if
can be written as product of an even (resp. odd) number of transpositions. A
trivial term of the determinant is a term of (1), , equal to zero. If is a square submatrix of a matrix , with entries in , and all the terms of the determinant of are trivial we say that is a trivial minor of . We say that is superregular if all its nontrivial minors are different from zero.Several notions of superregular matrices have appeared in different areas of mathematics and engineering having in common the specification of some properties regarding their minors Ando1987 ; CIM1998 ; Gan59a ; Pinkus2009 ; Roth1985 . In the context of coding theory these matrices have entries in a finite field and are important because they can be used to generate linear codes with good distance properties. A class of these matrices, which we will call full superregular, were first introduced in the context of block codes. A full superregular matrix is a matrix with all of its minors different from zero and therefore all of its entries nonzero. It is easy to see that a matrix is full superregular if and only if any linear combination of columns (or rows) has at most zero entries. For instance, Cauchy matrices are full superregular and can be used to build the socalled ReedSolomon block codes. Also, circulant Cauchy matrices can be used to construct MDS codes, see cl12 . It is wellknown that a systematic generator matrix generates a maximum distance separable (MDS) block code if and only if is full superregular, Roth1989 . The analog of Cauchy superregular matrices has been recently studied in depth in NERI2020 .
Convolutional codes are more involved than block codes and, for this reason, a more general class of superregular matrices had to be introduced.
Definition 1.1
(gl03, , Definition 3.3) A lower triangular matrix is defined to be superregular if all of its minors, with the property that all the entries in their diagonal come from the lower triangular part of , are nonsingular.
In this paper, we call such matrices LTsuperregular. Note that due to such a lower triangular configuration the remaining minors are necessarily zero. Roughly speaking, superregularity asks for all minors that are possibly nonzero, to be nonzero. In gl03 it was shown that Toeplitz LTsuperregular matrices can be used to construct convolutional codes of rate and degree that are strongly MDS provided that (for the rank analog of Toeplitz LTsuperregular matrices in the context of the rank metric, see ALMEIDA2020 ). This is again due to the fact that the combination of columns of superregular matrices ensures the largest number of possible nonzero entries for any linear combination (for this particular lower triangular structure). In other words, it can be deduced from gl03 that a lower triangular matrix , the columns of , is LTsuperregular if and only if for any linear combination of columns of , with , then , where
is the Hamming weight of a vector
, i.e., its number or nonzero coordinates. For a similar result but for more general classes of superregular matrices, not necessarily lower triangular, see (al16, , Theorem 3.1).It is important to note that in this case due to this triangular configuration it is hard to come up with an algebraic construction of LTsuperregular matrices. There exist however two general constructions of these matrices ANP2013 ; gl03 although they need very large field sizes. In this paper we will be interested in finding Toeplitz LTsuperregular matrices over small finite prime fields. So, our matrices will be of the form
(2) 
One important question is how large a finite field must be in order that a superregular matrix of a given size can exist over that field. For example, there exists no LTsuperregular matrix of order over the field because all the entries in the lower triangular part of a superregular matrix must be nonzero, which means that in this case all such entries would have to be ; clearly, this does not result in a superregular matrix, since the lower left submatrix of order is singular. The size of the smallest finite field for which exists an LTsuperregular matrix of order can be seen in Table 1. For the smallest finite field for which exists a Toeplitz LTsuperregular matrix of order is still unknown, but in Hutchinson2008 , Hutchinson et al. obtained an upper bound for its size and in HaOs2018 the authors showed the existence of LTsuperregular matrices of size over the field . In (Hutchinson2008, , Conjecture 3.5) and in gl03 it was conjectured, based on several examples, that an LTsuperregular matrix of size exits over for . Recently, new upper bounds on the necessary field size for the existence of these matrices and other superregular matrices with different structure, were presented in Lieb2019 .
Since the work in Hutchinson2008 is the motivation for this paper, we will give a brief description of their method to derive an upper bound on the minimum size a finite field must have in order that a superregular matrix of a given size can exist over that field.
Consider
a lower triangular Toeplitz matrix with indeterminate entries . The determinants of the proper square submatrices of such a matrix are given by nonzero polynomials in these indeterminates. Notice that in any of these polynomials at most the first power of can appear; i.e., each of these polynomials either it is linear in or does not appear in any of its terms. We study now those proper square submatrices of whose determinants are linear in . Denote by the set of such submatrices and by , the subset of formed by the submatrices of which are symmetric over the antidiagonal. Hutchinson et al. proved that is an upper bound for the number of different polynomials that can appear as the determinants of elements of . By computer search we found that is actually the exact number of such polynomials, for and , but for we have different polynomials and for we have different polynomials, whereas and . In Hutchinson2008 it is also proved that
(3) 
Therefore, given , a field and a lower triangular Toeplitz matrix (as in (2)), then has at most different minors that depend on the entry , all of them being linear on .
Remark 1.2
Notice that is an increasing sequence (and ). If we choose a field , such that , then we may choose such that , then select such that all the minors involving in the matrix
are nonzero (i.e. any ), then again we can choose such that all the minors involving in the matrix
are nonzero, and continuing in this way, we may eventually choose such that all of the minors involving in the matrix (as in (2)) are nonzero. Therefore, all the non trivial minors of this matrix just constructed, are nonzero. Therefore is LTsuperregular. This is the idea of the proof of Theorem 1.3.
Theorem 1.3
Hutchinson2008 Let be a finite field such that , then there exists a LTsuperregular matrix over .
Unfortunately this upper bound for the minimum field size is not very sharp, as Table 1 (obtained in Hutchinson2008 ) demonstrates. The actual minimum field sizes display in the table were obtained by randomised computer search.
Minimum Field size  Upper Bound ()  

3  3  3 
4  5  5 
5  7  11 
6  11  27 
7  17  77 
8  31  233 
9  59  751 
10  127  2495 
In this paper we continue the work initiated in [8] and study lower triangular Toeplitz superregular matrices over , an odd prime number. In particular, we investigate the number of different nonzero minors of these matrices. We show that this number is, in many cases, significantly smaller than the derived in Hutchinson2008 and therefore this immediately improves the upper bound given in Theorem 1.3 for the minimum field size necessary for the existence of this class of superregular matrices.
2 Smallest number of different nonzero minors of an LTsuperregular Toeplitz matrix
Since the multiplication by a constant does not change the superregularity of a matrix, we may assume that . The following lemma implies that we can also assume that .
Lemma 2.1
From now on, we will consider where is an odd prime number and
(4) 
In this section, we are interested in studying the smallest possible number of different minors for each with . For some of the values of we are able to compute the smallest number of different nonzero minors for every finite prime field. We will also exhibit plenty of superregular matrices for each .
2.1
If then there are two minors with the entry , namely and , so
(5) 
Hence must have at least elements. Therefore . For example, if , or , then is LTsuperregular.
2.2
If then , i.e., there are four different minors with the entry , on the variables and , namely
So, we must have
(6) 
Notice that if we put then there are only three different minors involving . Hence, in this case we can always choose, for example, . Therefore, we just proved the following result.
Theorem 2.2
For and , the number of different minors of involving is , and if then is LTsuperregular.
2.3
Although it is possible to construct LTsuperregular matrices over , with as the number of different minors involving can be reduced to by selecting properly and , and to for all . The ten different minors in the variables and are
Now, if is sufficiently large such that satisfies (5), satisfies (6) and
(7) 
then is LTsuperregular.
Notice that if and then
i.e.
and
In the case we even have . Also, if and then
i.e.
and
In this case, we also have . Moreover, it is easy to see that if then in both cases.
If we follow the previous subsections and consider and , we obtain
so there are ten different expressions involving for . Nevertheless, we always have . Therefore we have the following result.
Theorem 2.3
Let and .

If then has different minors involving ;

If then has different minors involving ;

If then is LTsuperregular;

If then is LTsuperregular.
2.4
We have and if is sufficiently large, and
(8) 
then all of the minors involving are nonzero (there are at most minors). We will show that for any prime we can choose and such that the number of elements of is at most (being smaller than for of the primes, because it will only be if and only if , as we will see below).
The following result about quadratic residues will be helpful.
Lemma 2.4
Let be an odd prime number. Then

is solvable if and only if ;

is solvable if and only if ;

is solvable if and only if ;

is solvable if and only if ;

is solvable if and only if ;
Proof:
and are well known results. Using the quadratic reciprocity law we easily obtain the remaining statements.
We are able to state and formally prove a result about the minimum number of elements of for any (computer search using Maple, helped us identify the necessary conditions).
Theorem 2.5
Let , and for each denote by one of the two solutions of , when they exist. Then

If and then , except when or , in which case we have ;

If and then , except when , in which case we have ;

If and then , except when , in which case we have ;

If and then , except when , in which case we have ;

If and then , except when , in which case we have , when , in which case we have and or in which case we have .
Moreover, if is any of the vectors above for an appropriate prime , except when and , then is LTsuperregular, for any .
Proof: For each prime , and using Maple, we found for which values of we would achieve the minimum of and after identifying which elements of become equal, we deduce the expressions stated in the theorem for . For each prime , there is at least one vector with for which is minimal. So we assume . Although, we use Maple to deduce expressions for and , the following arguments are valid, for every .
Suppose , with the calculations in Maple we found that is minimal when the thirteenth element of is null and the forth and twenty third elements are equal, i.e.
Solving this system of equations, we obtain
Substituting the first solution in , we obtain
Notice that has at most elements, for every prime for which exists. If and we take then and if we take then , it can be seen that for we also have , so the statement is obtained. Notice that the second solution is also in statement , since the solutions of are symmetric.
Suppose , with the calculations in Maple we found that is minimal when the thirteenth element of (in the expression (8)) is null and the third and fifth elements are equal, i.e.
Solving this system of equations, we obtain
Substituting the first solution in , we obtain
Here, has also at most elements, for every prime for which exists. If and we consider then and if we consider , we also obtain , hence the statement is proved. Notice that the second solution is also in statement , since the solutions of are symmetric.
Suppose with the calculations in Maple we found that is minimal when the thirteenth element of (in the expression (8)) is null and the fourth and tenth elements are equal, i.e.
Solving this system of equations, we obtain
but from Theorem 2.2 we cannot have . Substituting the first solution in , we obtain
Again, has at most elements, for every prime for which exists. As before, it can be seen that if then and so, we obtain the statement . Notice that the second solution is also in statement , since the solutions of are symmetric.
Suppose , with the calculations in Maple we found that is minimal when the thirteenth element of (in the expression (8)) is null and the third and sixth elements are equal, i.e.
Solving this system of equations, we obtain
Substituting the first solution in , we obtain
So, has at most elements, for every prime for which exists. Clearly, if then , but it can be seen that all the elements of are in . So, statement is obtained. Notice that the second solution is also in statement , since the solutions of are symmetric.
Suppose , with the calculations in Maple we found that is minimal when the eighth and the thirteenth elements of (in the expression (8)) are null, i.e.
Solving this system of equations, we obtain
Substituting the first solution in , we obtain
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