The notion of Generalized Hamming Weights (GHW), introduced by Wei in , is a generalization of minimum Hamming weight of a linear code. In  , the basic properties of GHW are studied and the weight hierarchy for Hamming code, Reed-Solomon codes, binary Reed-Muller code etc are determined.
Study of this notion was motivated by applications in cryptography. For instance, when a linear code is used over wire-tap channel of type II (see ), the amount of information revealed can be completely characterized using GHW hierarchy of the linear code. In a similar way, GHW can be used to analyze the performance of a linear code when used as a -resilient function . Later, study of GHW hierarchies found applications in determining the optimum bit order in trellis based decoding. Specifically, the GHW of RM codes found in  were used in  to prove that standard binary bit order is optimal for RM codes.
The -th GHW of a code is given by,
denotes the union of support of all the vectors in.
A geometric approach to determine GHW hierarchy for various classes of codes is described in . The projective Reed-Muller (PRM) codes, introduced by Gilles Lachaud in , are a variant of the Reed-Muller (RM) codes. These codes are based on evaluations of homogeneous polynomials of degree in projective space . The dimension and minimum distance of PRM codes were determined by Serre  and Srensen . In , Boguslavsky determined the second GHW of projective Reed-Muller codes for regime. The connections between Tsfasman-Boguslavsky conjecture and GHW of PRM codes were studied in . To the best of our knowledge, none of the previous works on GHW of PRM codes have considered the binary () version. However, next-to-minimal weight of binary PRM codes is determined in a recent work , wherein next-to-minimal weight means minimal codeword weight that is greater than minimum Hamming weight. Note that the next-to-minimal weight is not the same as second generalized Hamming weight.
In this paper we present the GHW hierarchy for binary PRM codes. In a recent work , by authors of the current paper, it was shown that binary PRM codes and their shortened versions have Private Information Retrieval (PIR) code property. The shortening technique proposed in that work resulted in an upper bound on the GHW of binary PRM codes. The work presented in this paper started as an attempt to prove the optimality of that shortening procedure. Here we derive a lower bound for the GHW of PRM codes and show that it matches with the upper bound provided in . The proofs presented in this paper adapt the ideas from derivation of GHW for RM codes in .
Organization of paper
In Section II we describe the parameters and properties of binary, projective Reed-Muller code. Section III discusses about the shortening procedure proposed in  that gives an upper bound on GHW of binary PRM codes. A lower bound is derived in Section IV using techniques from . In Section V we show that these bounds match and thereby determine the GHW hierarchy of the binary projective Reed-Muller code.
We use to denote the -th GHW for PRM code. The notation denotes and . The support of any code is denoted by .
Ii Binary Projective Reed-Muller Codes
Every codeword in the code over the field is a vector of evaluations of a homogeneous polynomial of degree at a fixed representative of each of the points in the projective space .
For , each point in the projective space has a unique representative with components, i.e, A codeword in binary PRM code is the vector of evaluations at all non-zero points in of a binary homogeneous polynomial of degree in variables.
where . The coefficients of monomials represents the message symbols. It can be easily seen that binary PRM code is a systematic code. Here we will be discussing only about binary PRM codes and hence from now on-wards PRM would mean the binary version.
Any binary homogeneous polynomial of degree evaluates to at vectors (in ) with Hamming weight less than . Hence, there are some coordinates which are always zero in all the codewords and can be deleted from the binary code. The non-degenerate code thus obtained has parameters:
Note that each codeword of this code is of the form , with . Any vector can be represented uniquely by its support. This implies that each code symbol can be indexed by a subset of with size . For an example code with , code symbol can be represented as where is the support of vector .
Each message symbol as well as its corresponding monomial can be indexed by a -element subset of .
Iii Shortening Algorithm : Upper Bound
In this section, we briefly describe the shortening technique proposed in , that resulted in upper bound on GHW of binary PRM codes.
For the code, any code-symbol , is given by:
where are the -element subsets of and are the message symbols.
For an example code with and , .
Consider that we set the message symbols , . This is equivalent to setting , because the code is systematic. Now we have,
This means that the coordinates corresponding to can be ignored. Hence, on shortening the PRM code by setting all message symbols corresponding to some -element subsets to zero, we can ignore the code coordinates corresponding to the message symbols and possibly some other code coordinates. Therefore, this shortening procedure will result in block length reduction of . The resultant code obtained will have parameters:
The aim of a good shortening algorithm for PRM code should be to pick these message symbols ( -element subsets of ) so that block length reduction is more. With this background, we state without proof the following lemmas from . For a given , and , first a unique vector is computed and then is computed using that. Note that here.
Lemma III.1 ( Unique representation ).
Any can be uniquely represented using a vector with and as,
Lemma III.2 ( Block length reduction  ).
Let be the unique representation of a given . Let , , be as defined in the previous lemma. By setting message symbols of PRM(r, m-1) code to zero, block length reduction of
The Table I shows the shortening procedure which results in Lemma III.2 for the case , . To reduce dimension by one has to pick first 2-element sets in the column and set corresponding message symbols to zero. For example, if , the message symbols given by , , and are set to zero.
The order in which the -element sets are picked here is called co-lexicographic order. For any two subsets and of an ordered set, we say in co-lexicographic order if , where . For instance we have, in co-lexicographic order since . Hence, , , , , , , , , , are in co-lexicographic order. Although it is not explicitly stated in , the general shortening procedure used to prove Lemma III.2 picks first element subsets of in co-lexicographic order.
The terminology anti-lexicographic order is used for the reverse co-lexicographic order. For example, , , , , , , , , , are in anti-lexicographic order. Hence, the remaining message symbols after shortening will correspond to the first element subsets of in anti-lexicographic order.
For the binary code, the -th generalized Hamming weight
where is the block length reduction given by Lemma III.2 for .
Proof: The shortened version of code obtained by setting first message symbols in co-lexicographical order to zero is a -dimensional sub code of the code. Therefore, the block length of this shortened code gives an upper bound on the -th GHW of the code.
Iv Lower Bound On GHW Of Binary PRM Codes
In Theorem IV.1 we present a lower bound on GHW for binary PRM codes. The proof shown here adapts techniques from the proof for Reed-Muller codes in . The GHW for RM codes determined in  give a lower bound on GHW for PRM codes since PRM codes are subcodes of RM codes with same parameters. However, we will prove that there is gap between GHW of RM and PRM codes (see Figure 1) by proving a tighter lower bound for PRM codes.
Every codeword in PRM code corresponds to evaluations of a binary homogeneous polynomial of degree in variables. Hence, we use the notation PRM to represent the codeword given by evaluations of homogeneous polynomial . It can be seen that any PRM can be represented as , where PRM and PRM.
For any ,
Proof: Let be a subcode of PRM with support size and dimension .
Let . We define,
Let be such that , where denotes direct sum, . We now define define
Let the dimension of , be , respectively. It can be observed that is a subcode of PRM and a subcode of PRM. Therefore, any element in can be written as , where are homogeneous polynomials and , .
We will now show that and have same dimension. If , then there exists such that . If there is such that , it would imply that . But , resulting in a contradiction. Therefore for every element in , there is a corresponding unique element in .
The support size of subcode is given by
where corresponds to support when for . Since is a dimensional subcode of PRM, we have and similarly .
It is clear to see that and . Thus we have,
V GHW Of Binary PRM Codes
In this Section, we will first state a well-known theorem from extremal set theory and then use it to prove the GHW results.
Let denote the family of all -element subsets of . For a collection , (upward) shadow is given by
The collection consisting of first -subsets of picked in anti-lexicographic order achieves and .
The collection consisting of first -subsets of picked in anti-lexicographic order achieves and .
Let denote the support size of subcode formed by monomials corresponding to first -element subsets of in the anti-lexicographic order. Note that this subcode is same as the shortened PRM code formed by setting message symbols picked in co-lexicographic order to zero. Since this subcode is dimensional subcode of PRM we have .
Proof: Suppose be the values that achieve minimum on the RHS. Then RHS . The RHS corresponds to the support size of subcode formed by monomials each of degree , with of them containing and of them without . Now, from Cor V.2 we know that picking the first -element sets in anti-lexicographic order results in minimum support size . Hence, RHS .
For any ,
The support of subcode formed by first monomials in anti-lexicographic order is the th GHW of PRM.
Proof: For the case of , the statement trivially follows. Now, we use induction over to prove the theorem and hence assume it is true for .
Let be the subcode with rank and support size . Then by Theorem IV.1 we have,
For any ,
where is obtained from Theorem III.2
Proof: From Theorem V.4, support of monomials picked in anti-lexicographic order gives the -th generalized Hamming weight. This is same as avoiding (shortening) the first , monomials picked in co-lexicographic order. Therefore, the support size obtained from Theorem III.2 with is equal to and hence the result.
The above corollary proves the optimality of shortening procedure for PRM codes given in  and provides the complete GHW hierarchy. GHW hierarchy of PRM codes for some parameters are listed in Table II. The next corollary gives simplified expression for GHW in some special cases.
Proof: Pick first , -element subsets of in anti-lexicographic order. These sets are of the form:
for all .
Now consider the monomials corresponding to , for all . The vector for which at-least one of these monomials evaluate to one has, for all and , for at-least one . The remaining can take any value. The number of such vectors is .
The GHW for PRM code obtained in Cor V.8 is by considering the sets to be removed. In the GHW derivation for Reed-Muller codes in , the counting is done taking into account the sets that remain. The following lemmas gives an expression for GHW of PRM codes using a similar approach.
Any can be uniquely represented by canonical form given by
where, , and .
Proof: We induct over variable .
For , the result is trivial. Assume that the statement true for .
If , define . Then, has a canonical representation given by:
Setting , and and for all satisfies the lemma statement with .
For , the canonical form will itself be the canonical form.
For any , the k-th GHW of binary projective Reed Muller code PRM is given by:
Proof: Here, we induct on variable . Assume that the result holds for the case of .
Let be the set of first , r-element subsets of in anti-lexicographic order.
Consider the case of , here exhausts all the -element subsets that include . Suppose represents support generated by sets that include and the support generated by remaining sets in , where . Then,
It can be observed that is same as the block length of PRM code. Therefore by induction assumption,
where is the canonical representation of .
Now by picking , , , for all and , we get:
The second equality follows from Theorem V.4.
For the case of , all the -element sets in include and the support generated by these sets can therefore be determined by . The canonical representation for is also canonical representation for .
Equation (8) follows by induction assumption.
-  M. Vajha, V. Ramkumar, and P. V. Kumar, “Binary, shortened projective reed muller codes for coded private information retrieval,” in IEEE International Symposium on Information Theory, ISIT, 2017.
-  V. K. Wei, “Generalized hamming weights for linear codes,” IEEE Trans. Information Theory, vol. 37, no. 5, pp. 1412–1418, 1991.
-  L. H. Ozarow and A. D. Wyner, “Wire-tap channel ii,” Bell Labs Technical Journal, vol. 63, no. 10, pp. 2135–2157, 1984.
-  B. Chor, O. Goldreich, J. Hasted, J. Freidmann, S. Rudich, and R. Smolensky, “The bit extraction problem or t-resilient functions,” in Foundations of Computer Science, 1985., 26th Annual Symposium on. IEEE, 1985, pp. 396–407.
-  T. Kasami, T. Takata, T. Fujiwara, and S. Lin, “On the optimum bit orders with respect to the state complexity of trellis diagrams for binary linear codes,” IEEE Transactions on Information Theory, vol. 39, no. 1, pp. 242–245, 1993.
-  M. A. Tsfasman and S. G. Vladut, “Geometric approach to higher weights,” IEEE Trans. Information Theory, vol. 41, no. 6, pp. 1564–1588, 1995.
-  G. Lachaud, “Projective reed - muller codes,” in Coding Theory and Applications, 2nd International Colloquium, 1986, pp. 125–129.
-  J.-P. Serre, “Lettre à m. tsfasman,” Astérisque, vol. 198, no. 200, pp. 351–353, 1991.
-  A. B. Sørensen, “Projective reed-muller codes,” IEEE Trans. Information Theory, vol. 37, no. 6, pp. 1567–1576, 1991.
-  M. Boguslavsky, “On the number of solutions of polynomial systems,” Finite fields and their applications, vol. 3, no. 4, pp. 287–299, 1997.
-  M. Datta and S. R. Ghorpade, “Remarks on tsfasman-boguslavsky conjecture and higher weights of projective reed-muller codes,” CoRR, vol. abs/1603.06232, 2016.
-  C. Carvalho and V. G. L. Neumann, “The next-to-minimal weights of binary projective reed-muller codes,” IEEE Trans. Information Theory, vol. 62, no. 11, pp. 6300–6303, 2016.
-  J. B. Kruskal, “The number of simplices in a complex,” Mathematical optimization techniques, vol. 10, pp. 251–278, 1963.
-  G. Katona, “A theorem of finite sets,” in Classic Papers in Combinatorics. Springer, 2009, pp. 381–401.