An code is a -ary linear code with length , dimension , and minimum Hamming distance . Since we will only consider binary codes, we also speak of codes. Linear codes have numerous applications so that constructions or non-existence results for specific parameters were the topic of many papers. One motivation was the determination of the smallest integer for which an code exists. As shown in  for every fixed dimension there exists an integer such that for all , where , is the so-called Griesmer bound. Thus, the determination of is a finite problem. In 2000 the determination of was completed in . Not many of the open cases for have been resolved since then and we only refer to most recent paper .
The aim of this note is to to circularize a recent application of non-existence results of linear codes. In random linear network coding so-called constant-dimension codes are used. These are sets of -dimensional subspaces of with subspace distance . By we denote the maximum possible cardinality, where , so that we assume . In  the upper bounds for and were proven. Here denotes the maximal integer such that there exists a -divisible -ary linear code of effective length and a code is called -divisible if the Hamming weights of all codewords are divisible by . For integers the possible length of -divisible codes have been completely determined in  and except for the cases and no tighter bound for with is known. For the case , where the constant-dimension codes are also called partial spreads, the notion of can be sharpened by requiring the existence of a projective -divisible -ary linear code of effective length . Doing so, all known upper bounds for follow from non-existence results of projective -divisible codes, see e.g. . For each field size and each integer there exists only a finite set such that there does not exist a projective -divisible code of effective length iff . We have , , and remark that the determination of was recently completed in  by excluding length .
In this paper we show the non-existence of -divisible binary codes of effective length , which e.g. implies .
Since the minimum Hamming distance is not relevant in our context, we speak of codes. The dual code of an code is the code consisting of the elements of that are perpendicular to all codewords of . By we denote the number of codewords of of weight . With this, the weight enumerator is given by . The numbers of codewords of the dual code of weight are related by the so-called MacWilliams identities
Clearly we have . In this paper we assume that all lengths are equal to the so-called effective length, i.e., . A linear code is called projective if . Let be a projective code. By comparing the coefficients of , , , and on both sides of Equation 1 we obtain:
The weight enumerator of a linear code can be refined to a so-called partition weight enumerator, see e.g. . To this end let be an integer and be a partition of the coordinates . By we denote a multi-index, where and for all . With this, denotes the number of codewords such that for all , which generalizes the notion of the counts . By we denote the corresponding counts for the dual code of . The generalized relation between the and the is given by:
The support of a codeword is the set of coordinates . The residual of a linear code with respect of a codeword is the restriction of the codewords of to those coordinates that are not in the support of , i.e., the resulting effective length is given by . If is a codeword of a -divisible -ary code , where , then the residual code with respect to is -divisible, see e.g. . The partition weight enumerator with respect to a codeword is given by Equation (6), where we choose , , and , so that restricting to the coordinates in gives the residual code.
Iii No projective -divisible binary linear code of length exists
Assume that is a projective -divisible code. Since for every codeword the residual code is -divisible and projective, we conclude from , see e.g. , that the possible non-zero weights of the codewords in are contained in . For codewords of weight the weight enumerator of the corresponding residual code can be uniquely determined:
([3, Lemma 24])
The weight enumerator of a projective -divisible binary linear code of (effective) length is given by , i.e., it is an -dimensional two-weight code.
Each projective -divisible code satisfies
, and .
First we exclude the case of dimension :
No projective -divisible code exists.
Proof. For the equations of Lemma 2 yield
for a projective -divisible code . Since and , we have , so that , , , and . Now consider a codeword of weight and the unique codeword of weight . In the residual code of the restriction of has weight or due to Lemma 1. In the latter case the codeword has weight , which cannot occur in a projective -divisible binary linear code of length . Thus, we have that gives another codeword of weight . However, since
is odd, this yields a contradiction and the codedoes not exist.
A projective -divisible binary linear code of length does not contain a codeword of weight or .
Proof. Let be an arbitrary codeword of weight (which indeed exists, see Lemma 2) and a codeword of weight or . We consider the residual code of with respect to the codeword . From Lemma 1 we conclude that the restriction of in has weight , , or . Since has a weight of at most , is the zero codeword of weight . In other words, we have . If denotes the set of codewords of weight in , then , with and .
Now let be the code generated by the elements in , i.e., the codewords of weight . By we denote the dimension of and by the dimension of . Since contains all codewords of weight and due to Lemma 2 we have
for . Since each generator matrix of contains a column that occurs at least times, i.e., the maximum column multiplicity is at least . If a row is appended to then the maximum column multiplicity can go down by a factor of at most the field size , i.e., in our situation. Thus, we have . Since Inequality 7) gives
we obtain a contradiction. Thus, we conclude .
No projective -divisible binary linear code of length exists.
We remark that some parts of our argument can be replaced using the partition weight enumerator from Equation (6). If we consider the partition weight enumerator with respect to a codeword of weight , then we have , , and . The possible indices where might be positive are given by , , , , , , , , , , and . Clearly, we have and . By considering the sums of a codeword with we conclude , , , and . From Lemma 1 we conclude , , and , where is the dimension of the code and a free parameter. Plugging into Equation (6) this gives for the coefficients of since . Using this equation automatically gives , , and . Since the coefficient of gives . Thus, we have and . The coefficient of then gives . For the non-negativity conditions force , so that , , , , , , and . It can be checked that all coefficients on the right hand side of Equation (6) are non-negative. implies , so that would be negative for .
Theorem 5 implies a few further results.
For we have .
Proof. Assume that is a set of -dimensional subspaces in
with pairwise trivial intersection. Then, the number of vectors inthat are disjoint to the vectors of the elements of is given by . Thus, by [3, Lemma 16], there exists a projective -divisible binary linear code of length , which contradicts Theorem 5.
The recursive upper bound for constant-dimension codes mentioned in the introduction implies:
We have , , and .
As an open problem we mention that the non-existence of a projective -divisible binary linear code of length would imply .
For , , and no projective -divisible code exists.
Proof. In [3, Theorem 12] it was proven that the length of a projective -divisible binary linear code either satisfies or can be written as for some non-negative integers and . Using , we note that . If , then is divisible by , so that . However, for we have – contradiction.
For , , and no projective -divisible code exists.
Proof. Due to Lemma 8 it suffices to consider . The case is given by Theorem 5. For we proof the statement by induction on . Assuming the existence of such a code, Equation (3) minus times Equation (2) yields
The residual code of a codeword of weight is projective, -divisible, and has length . If , then we can apply Lemma 8 to deduce . For the induction hypothesis gives . Since for the left hand side of Inequality (8) is non-positive – contradiction.
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