New Lower Bounds for Permutation Codes using Linear Block Codes
In this paper we prove new lower bounds for the maximal size of permutation codes by connecting the theory of permutation codes with the theory of linear block codes. More specifically, using the columns of a parity check matrix of an linear block code, we are able to prove the existence of a permutation code in the symmetric group of degree , having minimum distance at least and large cardinality. With our technique, we obtain new lower bounds for permutation codes that enhance the ones in the literature and provide asymptotic improvements in certain regimes of length and distance of the permutation code.
Permutation codes have been of great interest recently due to their applications (for example in powerline communications [ch04, co04]) and for their intrinsecal combinatorial interest [fr77, ga13, ji16, ta99, wa17]. Let us now briefly explain what permutation codes are. The symmetric group can be endowed with a metric defined as follows: if , then . An -permutation code is a subset of such that . The maximal size of an -permutation code has been studied widely in the literature. Very nice ideas to produce lower bounds appeared in [ga13, ji16, wa17], and they all improve asymptotically the famous Gilbert-Varshamov bound. In this paper we provide new lower bounds for . From a theoretical point of view, the paper connects the theory of permutation codes with the theory of linear block codes and converts the problem of extistence of permutation codes with certain parameters into existence problems for some linear block codes. From a practical perspective, our approach allows to produce improved bounds for many set of parameters . Moreover, for certain choices of regimes of and we actually beat asymptotically the best known bounds in [ji16, wa17]. The paper is structured as it follows.
Section 2 recaps the basic tools we need from coding theory and the theory of permutation codes.
Section 3 provides the technical heart of our proof, which gives the wanted connection between the theory of permutation codes and the theory of linear block codes.
In Section 4 we use the results of Section 3 together with results from the theory of Maximum Distance Separable (MDS) codes to provide two new lower bounds on permutation codes. The first (Theorem 4.5) beats the bounds in [ji16, wa17] whenever the next prime power larger than or equal to is smaller than the next prime larger than or equal to (in all the other cases it gives the same bound). The second one (Theorem 4.9) beats asymptotically [ji16, wa17] in the large distance regime.
In Section 5 we produce new bounds using Almost MDS codes that provide additional improvements of the bounds in [ji16, wa17] under the assumption that a linear code with certain parameters exists.
Finally, in Section 6 we compare the bounds we obtained in the paper with the previous bounds in the literature.
Conclusions are provided in Section 7.
In this section we recall the basic notions of linear codes endowed with the Hamming distance, and the theory of permutation codes.
2.1 Linear Block codes
Let be a prime power and denote by the field with elements. For a given positive integer we consider, the Hamming distance over , that is the map
defined by for . Moreover, the Hamming weight of a vector is the quantity
In this context, an code is a -dimensional subspace of equipped with the Hamming distance. The integer is the length and is called the dimension of . The minimum distance of is the integer defined by
In the following we will use the notation for a code of length , dimension and minimum distance .
The dual code of an code is the code
where denotes the standard inner product between two vectors in .
Two important matrices are related to an code . A generator matrix for is a matrix in whose rows are a basis for , i.e. . A parity check matrix for is a matrix such that .
From the definition, it is straightforward to verify that a matrix is a parity check matrix for an code if and only if it is a generator matrix for the dual code .
Let be an code, be a parity check matrix for and let be a positive integer. The following are equivalent.
Every columns of are linearly independent over .
Two codes and are said to be equivalent if there exists , such that
In terms of their generator matrices an, respectively, parity check matrices, we can see the following. If and are generator matrices for and respectively, then and are equivalent if and only if there exists permutation matrix and diagonal matrix such that . An analogous statement holds with their parity check matrices.
Let and be parity check matrices for two codes and respectively. Then, and are equivalent if and only if there exists a permutation matrix and a diagonal matrix such that .
Let be an linear code . If has a codeword of Hamming weight , then there exists an code equivalent to which has a parity check matrix whose first row is equal to .
A parity check matrix for is a generator matrix for . Let be a codeword of Hamming weight , and take as a generator matrix for a matrix whose first row is . Define the matrix . Therefore, the code whose parity check matrix is is equivalent to and the first row of is equal to . ∎
2.2 Permutation codes
Let be a positive integer and denote by the symmetric group on elements. On the group we consider the Hamming distance, that is defined for , as
A permutation code of length is a subset of endowed with the Hamming distance. The minimum distance of is the quantity
Let be the maximum cardinality that a permutation code of length and minimum distance can have. There are many known bounds on this quantity, that we now briefly recall.
Theorem 2.7 (Singleton-like bound).
A derangement of size is a permutation on elements with no fixed points. Let denote the number of derangements of size . The number of derangements of size is also known as the subfactorial, and it is well-known that
Theorem 2.8 (Sphere-packing bound).
Theorem 2.9 (Gilbert-Varshamov bound).
An improvement of the Gilbert-Varshamov bound, at least from an asimptotical point of view, was given in [ji16], whose proof relies on rational function fields theory. Another proof of the same result can be found in [wa17].
[ji16, Theorem 2][wa17, Theorem 13]. For every prime , for every ,
3 Bounding Permutation Codes Using Linear Block Codes
In this section we provide a general lower bound on the maximal size of a permutation code of given length and minimum distance . The bound in Theorem 3.1 is the technical heart of the paper from which the explicit bounds in the next sections will follow.
Let be a positive integer. For a given subset of the symmetric group , we denote by the maximum cardinality of a permutation code of minimum distance at least entirely contained in , i.e.
Note that, with this notation, . In the next proposition we use the convention that . For a set and an element we denote by the set . Clearly, if is a permutation code of minimum distance , then also is a permutation code of minimum distance .
Let be integers such that and . Let moreover be a prime power and be positive integers such that and . If there exists an code such that has a codeword of Hamming weight , then
Let be an code such that has a codeword of Hamming weight . By Lemma 2.5 we have an code with a parity check matrix whose first row is . Let be the -th column of and let with . We can write and define the map
Moreover, choose the subgroup of defined as
One can see that .
Let be a permutation code of minimum distance and cardinality . Consider the set of right cosets of , that is for some ’s in . Define the set
From this set, we consider the map
Assume and . Let be the subset of such that . Then
Since are linearly independent, it follows for every . Therefore, and are equal over the integers on all the ’s not in (because of their distance), and they are equal modulo on all the ’s in (since the are all distinct elements of and by the independence of the ’s). This forces in particular that for any . Since the equation holds for any , by relabeling with , we get that for all . This implies that and also, by construction, we have . Since and , we obtain . This shows that for every the preimage is a permutation code of minimum distance at least . Moreover, since has as first row, , where
Therefore, by generalized pigeonhole principle, we have that there exists such that has cardinality at least
In the rest of the paper we will apply Theorem 3.1, as we will be always able to show the existence of a codeword of weight in the dual of the code. Nevertheless, one can also show the following
Let be integers such that and . Let moreover be a prime power and be positive integers such that and . If there exists an code , then we have
The proof is completely analogous except for the fact that is not anymore included in (as does not necessarily has in the first row all ’s). Therefore, in the last step one simply has to replace with getting
4 Lower bounds using MDS codes
In this section we are going to apply the result of Theorem 3.1 using a specific class of linear codes, namely the MDS codes.
Theorem 4.1 (Singleton Bound [si64]).
Let be an code. Then
The Singleton defect of an code is the number . Observe that, by Theorem 4.1, the Singleton defect of a linear code is always a non-negative integer.
Recall that, for fixed and , the lower bound on provided in Theorem 3.1 depends on the existence of an code , and it contains a factor in the denominator. Since , it is only useful to consider codes with small Singleton defect.
An code with is called maximum distance separable (MDS) code.
Whenever an -code is MDS, we write that is an MDS code.
MDS codes have been deeply studied over the last 60 years because of their optimal parameters [ma77, va12] and their connection to finite projective geometry [se55, br88]. In the following we recall few of their basic properties.
Let be an MDS code. Then is an MDS code.
[ez11, Theorem 6] Any MDS code with has a codeword of weight for every . In particular, for every , a code has codewords of weight .
For every and every MDS code , the dual code has a codeword of weight .
For every prime power , and every integer with ,
Theorem 4.6 provides a lower bound on , using the existence of MDS codes of length over a finite field with cardinality at least . The rest of the section is devoted to obtain a similar bound, using MDS codes whose length exceeds the cardinality of the underlying finite field.
[ez11, Theorem 8] A MDS code has a codeword of weight for every , except for the -ary symplex code , that has only codewords of weight and . In particular, for every , a code has codewords of weight .
For every and every MDS code , the dual code has a codeword of weight .
For every prime power , and every ,
5 A lower bound using Almost MDS codes
In Section 4 we have already studied the bound with respect to MDS codes, hence in this section we will deal with codes with Singleton defect equal to .
An code with is called Almost MDS (or AMDS for short).
Almost MDS codes have been deeply studied in literature, since they represent the closest family to the one of MDS codes. Some classical examples of those codes arise from algebraic-geometric codes obtained using curves of genus [ts13]. For the interested reader we refer to [de96, do95, fa97].
Let be a prime power, be three positive integers such that . If is an code with , then has a codeword of weight .
Consider a generator matrix for that, after permutation of coordinates, we can assume of the form . Then, a generator matrix for is given by . Since , the rows of are all non-identically zero. Indeed, if one of them were identically zero, then we would find a codeword of weight in . Take now an element . Then, is of the form , and we assume . In this way the last entries of are non-zero. Therefore, we want to prove that there exists such that also the first entries of are non-zero.
Let us call the -th row of , that is also the -th column of . Let us define the sets
so that all the first entries of are non zero. We can give an estimation on the sets as follows. We observe that every is described by zeros of a linear polynomial in variables. By Schwartz-Zippel Lemma [sc79, Lemma 1] we have , and hence . Since , we conclude observing that
In Section 3, we have introduced the function for any positive integer and any subgroup of some symmetric group. In the special case that is the direct product of copies of , we can associate the function to a very well-known function in coding theory.
Let be a prime power, and be two positive integers such that . We define the number as the maximum cardinality of a non-necessarily linear code of length and minimum distance over , i.e.
Let be a subgroup of the form . Then .
The subgroup can be seen as, after relabeling the elements , the subgroup
is a bijective homothety, i.e. it preserves the distance up to a scalar multiple. In fact, we have that for every
Therefore by the maximality of and by the maximality of . The claim follows as is a bijection. ∎
Let be two positive integers such that and be a prime power with . If there exists an AMDS code such that has a codeword of weight , then
Let be two positive integers such that and be a prime power with . If there exists an AMDS code , then
6 Comparison with the previous bounds
We explain here how our bounds compare with others given in the literature. As our Theorem 4.5 allows to be the next prime power greater or equal to , we beat (or at least equal) the bounds in [ji16, wa17] (see Table 1). Interestingly enough, when is a prime power, Theorem 4.8 beats asymptotically the bounds in [ji16, wa17] in the large distance regime. We formalize this in the proposition below. Let us denote by the function that sends an integer to the smallest prime number larger than or equal to , and by the function that sends an integer to the smallest prime number larger than or equal to .
For the rest of this section, we set
More specifically, represents the bound in [ji16, Theorem 2] and [wa17, Theorem 13], while and are the bounds in Theorem 4.6 and Theorem 4.9, respectively. It is trivial to see that , for every . We now focus on the comparisons of with and the bound given in Theorem 5.5.
Let , and set for some . Then,
In particular, for , gives asymptotically a better bound than .
It is important to show that in the regime where we beat the old bound, the new one is actually non-trivial. We do that in the following remark.
Observe that in the regime , the bound is asymptotically non-trivial. Indeed,
where the second inequality follows from Stirling’s approximation formula. Moreover, notice that the bound can only be used when is a prime power.
The following proposition shows the regime in which our bound in Theorem 5.6 beats by a large scale the previous known bounds.
|n||Theorem 4.6||Theorem 4.9||[ji16, wa17]|
|11||2 727||2 727|
|12||16 772||16 359||16 772|
|13||218 026||218 026|
|14||1 330 236||1 526 178||1 043 789|
|15||19 953 528||15 656 834|
|16||319 256 438||250 509 332|
|17||4 258 658 638||2 713 679 719||4 258 658 638|
|18||49 127 720 826||38 327 927 742||49 127 720 826|
|19||933 426 695 689||933 426 695 689|
|20||8 693 872 621 156||9 334 266 956 886||8 693 872 621 156|
|21||182 571 325 044 256||182 571 325 044 256|
Let be a prime power and for some such that . Set with , and
as goes to infinity.
Since we assumed , then , and in turn . ∎
Again, we notice in the next remark that in the regime where we beat the old bound, the new one is actually non-trivial.
Observe that in the regime and , with the bound is non-trivial. Indeed
for going to infinity, where the second inequality follows from Stirling’s formula.
Proposition 6.3 shows that the bound given in Theorem 5.6 could beat by far the bound in [ji16, Theorem 2] and [wa17, Theorem 13], and therefore also the one from Theorem 4.6, for and large enough. The reader should notice that Proposition 6.3 is conditioned to the existence of a family of AMDS codes of length over , for large and fixed. The existence of such family is not proven nor disproven and explicit constructions of linear codes with these parameters becomes now central also in the theory of permutation codes.
In this paper we connected the theory of linear codes with the theory of permutation codes. In turn, this allows to produce new lower bounds for the maximal size of permutation codes. The lower bounds produced use the existence of certain codes of given distance and length over an alphabet of a given size, converting the problem of finding a lower bound for permutation codes of given distance into the problem of finding a certain linear codes with parameters as in Theorem 3.1. In Section 6 we apply Theorem 3.1 and obtain improved bounds with respect to the ones in the literature [ji16, wa17], as one can now select a the next prime power instead of the next prime in the bound of [ji16, Theorem 2] and [wa17, Theorem 13] (thanks to our Theorem 4.6). Moreover, in Proposition 6.1 and Proposition 6.3 we show that we beat them asymptotically for certain regimes of and .