LET Õ Ò µ denote the maximum size of a Õ-ary code

1 Long Nonbinary Codes Exceeding the Gilbert-Varshamov bound for Any Fixed Distance Sergey Yekhanin Ilya Dumer Abstract Let Õ µ denote the maximum size of a Õ- ary code of length and distance We study the minimum asymptotic redundancy Õ µ ÐÓ Õ Õ µ as grows while Õ and are fixed For any and Õ long algebraic codes are designed that improve on the BCH codes and have the lowest asymptotic redundancy Õ µ º µ ¾µµ ÐÓ Õ known to date Prior to this work, codes of fixed distance that asymptotically surpass BCH codes and the Gilbert-Varshamov bound were designed only for distances and Index Terms affine lines, BCH code, Bezout s theorem, norm I INTRODUCTION LET Õ µ denote the maximum size of a Õ-ary code of length and distance We study the asymptotic size Õ µ if Õ and are fixed as, and introduce a related quantity ÐÓ Õ Õ µ Õ µ ÐÑ ÐÓ Õ which we call the redundancy coefficient The Hamming upper bound Õ µ Õ µ¾ leads to the lower bound Õ µ Õ µ µ¾ (1) which is the best bound on Õ µ known to date for arbitrary values of Õ and On the other hand, the Varshamov existence bound admits any linear Õ code of dimension ¾ ÐÓÕ Õ µ This leads to the redundancy coefficient ± Õ µ ¾ (2) (Note that the Gilbert bound results in a weaker inequality Õ µ ) S Yekhanin is a PhD student at the Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA His research was supported in part by NTT Award MIT 2001-04 and NSF grant CCR 0219218 I Dumer is with the College of Engineering, University of California, Riverside, CA 92521, USA His research was supported by NSF grant CCR 0097125 Let be a primitive element of the Galois field Õ Ñ Consider (see [20]) the narrow-sense BCH code defined by the generator polynomial with zeros ¾ Let Õ Ñ µ denote the extended BCH code obtained by adding the overall parity check Code Õ Ñ µ has length ÕÑ constructive distance, and redundancy coefficient ¾µ Õ µ Õ µ (3) Õ Note that the above BCH bound (3) is better than the Varshamov bound (2) for Õ and coincides with (2) for Õ Note also that (3) meets the Hamming bound (1) if Õ ¾ or Therefore ¾ µ µ¾ and Õ µ For distances and infinite families of nonbinary linear codes are constructed in [5] and [6] that reduce asymptotic redundancy (3) Open Problem 2 from [6] also raises the question if the BCH bound (3) can be improved for larger values of Our main result is an algebraic construction of codes that gives an affirmative answer to this problem for all Õ In terms of redundancy, the new bound is expressed by Theorem 1 : For all Õ and Õ µ µ ¾µ (4) Combining (3) and (4), we obtain ¾µ Õ µ Õ µ Ñ Õ µ ¾µ Note that the above bound is better than the Varshamov existence bound for arbitrary Õ and The rest of the paper is organized as follows In Section II, we review the upper bounds for Õ µ that surpass the BCH bound (3) for small values of In Section III, we present our code construction and prove the new bound (4) This proof rests on important Theorem 4, which is proven in Section IV Finally, we make some concluding remarks in section V II PREVIOUS WORK Prior to this work, codes that asymptotically exceed the BCH bound (3) were known only for We start with the bounds for Õ µ Linear Õ codes are equivalent to caps in projective geometries Õµ and have been studied extensively under this name See [18] for a review However, the exact values of Õ µ remain unknown for all Õ and the gaps between the upper and the lower bounds are still large

2 The Hamming bound yields Õ µ Mukhopadhyay [22] obtained the upper bound Õ µ For all values of Õ this was later improved by Edel and Bierbrauer [7] to Õ µ ÐÓ Õ Õ Õ ¾ µ (5) Note that for large values of Õ the right hand side of (5) tends to The case of Õ has been of special interest, and general bound (5) has been improved in a few papers (see [17], [11], [2], [8]) The current record µ due to Edel [8] slightly improves on the previous record µ obtained by Calderbank and Fishburn [2] For Õ the construction of [14] also improves (5) Namely, µ Now we proceed to the bounds for Õ µ The Hamming bound yields Õ µ ¾ Several families of linear codes constructed in [6] reach the bound Õ µ (6) for all values of Õ Later, alternative constructions of codes with the same asymptotic redundancy were considered in [9] Similarly to the case of there exist better bounds for small alphabets Namely, Goppa pointed out that ternary double error-correcting BCH codes asymptotically meet the Hamming bound (1) For Õ and two different constructions that asymptotically meet the Hamming bound were proposed in [12] and [4] Thus, µ µ ¾ For the infinite families of linear codes designed in [5] and [6] reach the upper bound Õ µ (7) for all Õ The constructions are rather complex and the resulting linear codes are not cyclic Later, a simpler construction of a cyclic code with the same asymptotic redundancy was proposed in [3] Again, better bounds exist for small values of Õ Namely, µ ¾ [6] and µ [10] We summarize the bounds described so far in Figure 1 The following Lemma 2 due to Gevorkyan [13] shows that redundancy Õ µ cannot increase when the alphabet size is reduced Lemma 2 : For arbitrary value of distance, Õ Õ ¾ µ Õ µ Õ ¾ µ Proof: Given a code Î of length over the Õ ¾ -ary alphabet we prove the existence of a code Î of the same length over Õ - ary alphabet with the same redundancy coefficient Let Õ ¾ -ary alphabet be an additive group Õ¾, and Õ -ary alphabet form a subset Õ Õ¾ Define the componentwise shift Î Ú Î Ú of code Î by an arbitrary vector Ú ¾ Õ ¾ Note that any vector ¾ Õ belongs to exactly Î codes among all Õ ¾ codes Î Ú, as Ú runs through Õ ¾ Hence, codes Î Ú include on average Õ Î Õ ¾ vectors of the subset Õ Õ ¾ Therefore, some distance d 11 10 9 8 7 6 5 4 3 2 1 2 3 4 5 6 7 8 9 10 11 alphabet q Fig 1 Hamming bound is tight Edel cap c(3,4) Glynn Tatau cap c(4,4) Edel Bierbrauer bound for c(q,4) Bound for c(4,6) by Feng et al Dumer bounds for c(q,5) and c(q,6) No bound better than BCH bound is known New bound A taxonomy of best known upper bounds for Õ µ set Î Ú Õ has at least this average size Denote this set by Î Clearly, Î is a Õ -ary code with the same distance as code Î It remains to note that ÐÓ Õ Õ Î Õ ¾ µ ÐÓ Õ ÐÓ Õ ¾ Î ÐÓ Õ¾ The proof is completed Corollary 3 : Let Õ be an infinite sequence of growing alphabet sizes Assume there exist and such that for all, Õ µ Then Õ µ for all values of Õ Proof: This follows trivially from Lemma 2 III CODE CONSTRUCTION In the sequel, the elements of the field Õ are denoted by Greek letters, while the elements of extension fields Õ are denoted by Latin letters We start with an extended BCH code Õ Ñ µ of length Õ Ñ and constructive distance Here for any position ¾ Õ Ñ we define its locator, where for and Then the parity check matrix of code has the form À Ñ Õ µ (8) Here the powers of locators are represented with respect to some basis of Õ Ñ over Õ Note that the redundancy of is at most µñ Also, we assume in the sequel that Õ does not divide ¾ since code has constructive distance instead of otherwise Consider any nonzero codeword ¾ of weight Û with nonzero symbols in positions Û Let µ Ü Ü Û denote its locator set, where we use notation Ü for all Û We say that µ lies on an affine line Ä µ over Õ if there exist ¾ Õ Ñ such that Ü (9)

3 where ¾ Õ for all values of Û The key observation underlining our code construction is that under some restrictions on extension Ñ and characteristic char Õ of the field Õ any code vector ¾ of weight has its locator set µ lying on some affine line 1 Formally, this is expressed by Theorem 4 : Let Ñ be a prime, Ñ µ and char Õ Consider the extended BCH code Õ Ñ µ of constructive distance Then any codeword of minimum weight has its locator set µ lying on some affine line Ä µ over Õ We defer the proof of Theorem 4 till section IV and proceed with the code construction Let Ñ ¾µ ¾µ Consider the field Õ and its subfield Õ Let be the basis of Õ over Õ such that Õ is spanned by Let Ñ be an arbitrary basis of Õ Ñ over Õ Below we map each element Ü Ñ «of the field Õ Ñ onto the element Ü Ñ «(10) of the field Õ It is readily seen that for arbitrary ¾ Õ Ñ and ¾ Õ Recall that the norm [15] of Ü ¾ Õ (11) Æ Õ» Õ Üµ Æ Õ µ Õ ¾ ܵ Ü is a classical mapping from Õ to Õ (12) Now we are ready to present our code construction Consider the Õ-ary code µ of length Õ Ñ with the parity check matrix À Ñ Õ Æ ¾ µ Æ ¾ µ (13) where the locators and their powers are represented in Õ with respect to the basis and values of Æ ¾ are represented in Õ with respect to Recall that Æ ¾ ܵ takes values in Õ Therefore the redundancy of does not exceed µñ Below is the main theorem of the paper Theorem 5 : Suppose Ñ µ is a prime, and char Õ ; then code µ defined by (13) has parameters Õ Ñ Õ Ñ µñ Ñ ¾µ Õ 1 We shall also see that code Õ Ñ µ does have minimum distance under these restrictions Proof: Note that, since is a subcode of the extended BCH code defined in (8) Let be the set of all codewords of weight exactly It remains to prove that Assume the converse Let ¾ be a codeword of weight with locator set µ Ü Ü µ This implies that for some nonzero symbols ¾ Õ Ü Ø Ø Æ ¾ Ü µ (14) Note that ¾ Therefore according to Theorem 4, there exist from Õ Ñ and pairwise distinct ¾ Õ such that Ü Consider the affine permutation ܵ Ü of the entire locator set Õ Ñ, where and Clearly, maps each Ü onto, ie Ü It is well known ([1], [20]) that the extended BCH code is invariant under any affine permutation of the locators, so that is also a locator set in Indeed, for any Ø ¾ we have an equality Ø Ü µ Ø Ø Ø Ø Ü (15) We shall now demonstrate that (14) yields one more equation ¾ (16) Indeed, we use (11) and (12) to obtain Õ µ Õ Æ ¾ ÕØ ÕØ (17) Ø ¾ Ø Ø µ Ø where Ø are some polynomials in and Now the last equation in (14) can be rewritten as ¾ ¾ Ø µ Ø Ø µ Ø Ø Ø This gives (16), due to the two facts: Ø for all Ø according to (15) ¾ µ Æ ¾ µ and is nonzero, since is nonzero and norm is a degree Equations (15) and (16) form the linear system Ø Ø ¾

4 in variables with the Vandermonde matrix Ø µ Recall that are nonzero and are pairwise distinct Therefore these equations hold only if simultaneously Thus, our initial assumption that has weight does not hold This completes the proof Lemma 6 : Suppose char Õ ; then Õ µ µ ¾µ Proof: We estimate the asymptotic redundancy of the family of codes presented in Theorem 5 Here Õ and are fixed, while Ñ µ runs to infinity over primes Then µñ Ñ ¾µ Õ µ ÐÑ Ñ Ñ µ ¾µ (18) The proof is completed It is obvious that for every there exists an infinite family Õ of growing alphabets such that char Õ Combining Lemma 6 with Corollary 3, we get Theorem 1 The proof is completed To conclude, we would like to note that our construction of code (13) generalizes the construction of nonbinary double error-correcting codes from Theorem 7 in [6] IV AFFINE LINES Before we proceed to the proof of Theorem 4, let us introduce some standard concepts and theorems of algebraic geometry Let be an algebraically closed field and Ö Ø be two positive integers Let Ö ¾ Ü Ü Ø For any Ü Ø µ ¾ Ø, the matrix Â Ü Ö µ Ü Ü Ö Ü Ü Ü Ø Ü Ö Ü Ø Ü (19) is called the Jacobian of functions at point Ü The set Î of common roots to the system of equations Ü Ü Ø µ Ö Ü Ü Ø µ (20) is called an affine variety The ideal Á Î µ is the set of all polynomials ¾ Ü Ü Ø such that ܵ for all Ü ¾ Î One important characteristic of a variety is its dimension Ñ Î Dimension of a non-empty variety is a non-negative integer Let Ü Ø µ ¾ Î be an arbitrary point on Î The dimension of a variety Î at a point Ü, denoted Ñ Ü Î, is the maximum dimension of an irreducible component of Î containing Ü A point Ü ¾ Î such that Ñ Ü Î is called an isolated point We shall need the following lemma ([19], p166) Lemma 7 : Let Î be an affine variety with the ideal Á Î µ Ü Ü Ø Then for any Ü Ø µ ¾ Î and Ö ¾ Á Î µ rank Â Ü µ Ø Ñ Ü Î The next lemma is a corollary to the classical Bezout s theorem ([16], p53) Lemma 8 : Let Î be an affine variety defined by (20) Then the number of isolated points on Î does not exceed Ö Let Ø be fixed non-zero elements of some finite field Õ Consider a variety Î in the algebraic closure of Õ defined by the following system of equations Ü Ø Ü Ø Ø Ü ¾ Ø Ü ¾ Ø Ø Ü Ø Ø Ü Ø Ø Ø (21) Let Ü Ø µ be an arbitrary point on Î We say that Ü is an interesting point if for all Lemma 9 : Let Î be the variety defined by (21) Suppose char Õ Ø; then every interesting point on Î is isolated Proof: Let Ü Ø µ be an arbitrary interesting point on Î Let Ü Ü Ø µ denote the left hand side of the -th equation of (21) Consider the Jacobian of at point Ü Â Ü Ø µ Thus we have Ø Â Ü Ø µ Ø Ø ¾ ¾ Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Ø Using standard properties of the Vandermonde determinant and the facts that are non-zero and char Õ Ø, we get rank Â Ü Ø µ Ø (22) It is easy to see that Ø ¾ Á Î µ Combining (22) with Lemma 7, we obtain Ñ Ü Î The proof is completed Lemma 10 : Let Ñ be a prime Ñ Ø Assume char Õ Ø Let Î be the variety defined by (21) Suppose Ü ¾ Õ Ø Ñ is an interesting point on Î ; then Ü ¾ Õ Ø In other words, every interesting point on Î that is rational over Õ Ñ is rational over Õ Proof: Assume the converse Let Ü Ø µ be an interesting point on Î such that Ü ¾ Õ Ø Ñ Õ Ø Consider the following Ñ conjugate points Ô Õ Õ Ø µ for all Ñ Each of the above points is interesting Since Ñ is a prime, the points are pairwise distinct However, according to Lemma 9 every interesting point on Î is isolated Thus, we have Ñ Ø isolated point on Î This contradicts Lemma 8 Remark 11 : Note that we can slightly weaken the condition of Lemma 10 replacing Ñ prime and Ñ Ø

5 with condition:, Ñ implies Ø Now we are ready to prove Theorem 4 Proof: Assume is nonempty (this fact will be proven later) and consider the locator set µ Ü Ü µ for any ¾ Recall that µ satisfies the first ¾ equations in (14) where for all Consider an affine permutation ܵ Ü of the locator set Õ Ñ of the code Let ¾ Õ Ñ be such that Ü ¾ µ and Ü µ (23) Let Ý denote Ü µ Now we again use the fact that code (8) is invariant under affine permutations Therefore the new locator set Ý µ Ý Ý µ satisfies similar equations ¾ Ý Ý ¾ Ý ¾ Ý ¾ ¾ Ý Ý ¾ (24) Now we remove the first equation (which does not include variables Ý µ from (24), and obtain the system of equations, which is identical to system (21) for Ø Recall that Ü Ü are pairwise distinct elements of Õ Ñ Therefore Ý Ý are also pairwise distinct Thus Ý Ý is an interesting solution to the above system It is straightforward to verify that all the conditions of Lemma 10 hold This yields Ý Ü ¾ Õ ¾ Thus, we obtain all locators Ü on the affine line Ü ¾ Õ Finally, we prove that is nonempty Note that char Õ ¾ Also, recall that we consider codes Õ Ñ µ with constructive distance in which case Õ does not divide ¾ Thus, we now assume that Õ Then we consider (24) taking and arbitrarily choosing different locators Ý Ý from Õ Obviously, the resulting system of linear equations has nonzero solution ¾ This gives the codeword of weight and completes the proof of Theorem 4 V CONCLUSION We have constructed an infinite family of nonbinary codes that reduce the asymptotic redundancy of BCH codes for any given alphabet size Õ and distance if Õ Families with such a property were earlier known only for distances, and [6] Even the shortest codes in our family have very big length Õ µ, therefore the construction is of theoretical interest The main question (ie the determination of the exact values of Õ µ) remains open ACKNOWLEDGEMENT S Yekhanin would like to express his deep gratitude to M Sudan for introducing the problem to him and many helpful discussions during this work He would also like to thank J Kelner for valuable advice REFERENCES [1] R Blahut, Algebraic Codes for Data Transmission Cambridge: Cambridge University Press, 2003 [2] AR Calderbank, PC Fishburn, Maximal Three-Independent Subsets of ¾, Designs, Codes and Cryptography, vol 4, pp 203-211, 1994 [3] D Danev, J Olsson, On a Sequence of Cyclic Codes with Minimum Distance Six, IEEE Trans on Inform Theory, vol 46, pp 673-674, March 2000 [4] II Dumer and VV Zinoviev, New maximal codes over Galois field GF(4), Probl Peredach Inform (Probl Inform Transm), vol 14, no 3, pp 24-34, 1978 [5] II Dumer, Nonbinary codes with distances 4,5 and 6 of cardinality greater than the BCH codes, Probl Peredach Inform (Probl Inform Transm), vol 24, no 3, pp 42-54, 1988 [6] II Dumer, Nonbinary double error-correcting codes designed by means of algebraic varieties, IEEE Trans on Inform Theory, vol 41, pp 1657-1666, November 1995 [7] Y Edel, J Bierbrauer, Recursive Constructions for Large Caps, Bulletein of Belgian Mathematical Society - 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