Optimal Ternary Cyclic Codes From Monomials

Transcription

1 5898 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 59, NO 9, SEPTEMBER 2013 Optimal Ternary Cyclic Codes From Monomials Cunsheng Ding, Senior Member, IEEE, and Tor Helleseth, Fellow, IEEE Abstract Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms Perfect nonlinear monomials were employed to construct optimal ternary cyclic codes with by Carlet, Ding, and Yuan in 2005 In this paper, almost perfect nonlinear monomials, and a number of other monomials over are used to construct optimal ternary cyclic codes with the same Nine open problems on such codes are also presented Index Terms Almost perfect nonlinear (APN) functions, cyclic codes, monomials, perfect nonlinear functions, planar functions I INTRODUCTION L ET be a power of a prime A linear code over is a -dimensional subspace of with minimum (Hamming) nonzero weight A linear code over the finite field is called cyclic implies Let By identying any vector with any code of length over corresponds to a subset of The linear code is cyclic and only the corresponding subset in is an ideal of the ring It is well known that every ideal of is principal Let be a cyclic code, where is monic and has the least degree Then is called the generator polynomial and is referred to as the parity-check polynomial of The error correcting capability of cyclic codes may not be as good as some other linear codes in general However, cyclic codes have wide applications in storage and communication systems because they have efficient encoding and decoding algorithms [10], [17], [25] For example, Reed Solomon codes have found important applications from deep-space communication to consumer electronics They are prominently used in consumer electronics such as CDs, DVDs, Blu-ray Discs, in data transmission technologies such as DSL & WiMAX, in broadcast Manuscript received November 28, 2012; revised April 13, 2013; accepted April 21, 2013 Date of publication April 30, 2013; date of current version August 14, 2013 C Ding was supported by The Hong Kong Research Grants Council Project T Helleseth was supported by the Norwegian Research Council C Ding is with the Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Kowloon, Hong Kong ( cding@usthk) T Helleseth is with the Department of Informatics, The University of Bergen, N-5020 Bergen, Norway ( TorHelleseth@iiuibno) Communicated by N Kashyap, Associate Editor for Coding Theory Digital Object Identier /TIT systems such as DVB and ATSC, and in computer applications such as RAID 6 systems Perfect nonlinear monomials were employed to construct optimal ternary cyclic codes with by Carlet et al [8] In this paper, almost perfect nonlinear (APN) monomials and a number of classes of monomials over will be used to construct many classes of optimal ternary cyclic codes with the same Nine open problems on such codes are also presented II PRELIMINARIES In this section, we fix some basic notation for this paper and introduce APN and planar functions, and -cyclotomic cosets that will be employed in subsequent sections A Some Notation Fixed Throughout This Paper Throughout this paper, we adopt the following notation unless otherwise stated: 1) is an odd prime, and for Section IV and subsequent sections 2) is a positive integer 3) 4) associated with the integer addition modulo and integer multiplication modulo operations 5) is a generator of 6) is the minimal polynomial of over 7) is the trace function from to 8) By the database, we mean the collection of the tables of best linear codes known maintained by Markus Grassl at B -Cyclotomic Cosets Modulo The -cyclotomic coset modulo containing is defined by where is the smallest positive integer such that, and is called the size of Itisknownthat divides The smallest integer in is called the coset leader of Let denote the set of all coset leaders By definition, we have The following lemma is useful in the sequel Lemma 21: Let and Forany with, the length of the -cyclotomic coset is equal to IEEE

2 DING AND HELLESETH: OPTIMAL TERNARY CYCLIC CODES FROM MONOMIALS 5899 Proof: By definition, is the smallest positive integer such that Hence, is the smallest positive integer such that Since, is the smallest positive integer such that It then follows that This completes the proof C Perfect and APN Functions on A function is called APN and is referred to as perfect nonlinear or planar C2: the equation has the only solution in ;and C3: the equation has the only solution in Proof: Clearly, the minimum distance of the code cannot be 1 The code has a codeword of Hamming weight 2 and only there exist two elements and in and two distinct elements and in such that Note that Wehavethat Hence, (1) is equivalent to the following set of equations: (1) There is no perfect nonlinear (planar) function on Known APN polynomials over can be found in [1] [5], [15], [16], [18] [20], and [24] However, there are both planar and APN functions over for odd primes Inthesequel, planar and APN monomials over will be employed to construct optimal ternary cyclic codes The results of this paper will be a nice demonstration of applications of APN functions in engineering III CODES DEFINED BY PAIRS OF MONOMIALS Let be a prime and let is a positive integer Let denote the minimal polynomial over of, where is a generator of In this paper, we consider the cyclic code of length over with generator polynomial, denoted by and The condition that is to make sure that and are distinct The dimension of is equal to and is the -cyclotomic coset modulo containing The cyclic code is defined by the pair of monomials and over, it was proved in [6], [7], and [23] that the binary code has and only is an APN monomial over is odd and is a planar monomial over, the codes were dealt with in [8] and [26] and,thecode has minimum distance 2 or 3 which may not be interesting In this paper, we study the case that only and prove that the ternary cyclic code is optimal for many classes of monomials over IV BASIC THEOREM ABOUT THE TERNARY CYCLIC CODES From now on, we consider only the code for We are mostly interested in the case that and In this case, the dimension of the code is equal to The following theorem is the fundamental result of this paper and will be used frequently in subsequent sections Theorem 41: Let and The cyclic code has and only the following conditions are satisfied: C1: is even; It then follows that does not have a codeword of Hamming weight two and only is even The code has a codeword of Hamming weight 3 and only there exist three elements,,and in and three distinct elements,,and in such that Due to symmetry, it is sufficient to consider the following two cases Case A, Where : In this case, (2) becomes Putting and Then, Hence, (3) has a desired solution and only the equation has a solution Case B, Where and : In this case, (2) becomes Putting and Then, Hence, (4) has a desired solution and only the equation has a solution By the Sphere Packing bound [21, Th 1121], the minimum distance is at most 4 Hence, Conditions C1, C2, and C3 are satisfied This completes the proof of this theorem Any ternary linear code with is optimal in the sense that the minimum distance is maximal for the fixed length and the fixed dimension In subsequent sections, we will present many classes of optimal ternary cyclic codes with these (2) (3) (4)

3 5900 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 59, NO 9, SEPTEMBER 2013 V OPTIMAL CYCLIC CODES DEFINED BY PLANAR MONOMIALS OVER We first prove the following lemma, where the first and the last conclusion should be known Lemma 51: If is APN or planar over,then must be even; ; ;and Proof: If is odd, then the equation will have three solutions and Bydefinition is not planar and APN This proves the first conclusion Let be planar or APN Then, must be even In this case, the equation has already one solution If,then In this case, the equation has at least four distinct solutions, 1,, and Define for all Then,,,and are three distinct solutions of the equation This is contrary to the assumption that is planar or APN Hence, We now prove the third conclusion Since, must divide Hence, Note that is odd for any and is even We have Hence, The following theorem was proved by Carlet et al in [8] For completeness, we report it here and present a proof for it Theorem 52 [8]: If is planar over,then is an optimal ternary cyclic code with Proof: Let be planar By Lemma 51, is even Notice that is already a solution of the equation Bythedefinition of planar functions, this equation cannot have other solutions Hence, Condition C3 in Theorem 41 is met We now prove that Condition C2 in Theorem 41 holds Suppose on the contrary that for some Then that is Note that as Thisiscontrarytotheassumption that is planar Therefore, Condition C2 in Theorem 41 is also satisfied By Lemma 51 and Lemma 21, the dimension of this code is equal to The desired conclusions then follow from Theorem 41 The following is a list of known planar monomials over : 1) 2) is odd ([12]) 3) and is odd ([11]) Each of these planar monomials gives a class of optimal ternary cyclic codes with More planar functions can be found in [11], [13], [27], and [28] Example 53: Let and let Let be the generator of with Then, is a ternary cyclic code with and generator polynomial The dual of is a ternary cyclic code with This dual code has the same as the best known ternary linear code with in the database The upper bound on the minimum distance of any ternary linear code of length 80 and dimension 8 is 49 The following theorem is the partial inverse conclusion of Theorem 52 when is of a special form Theorem 54: Let Then, the ternary cyclic code has andonly is odd and, ie, and only is planar over Proof: In the proof of Theorem 41, we see that has no codeword of Hamming weight 2 and only is even Hence, must be odd Notice that multiplying the left-hand sides of the two equations in Conditions C2 and C3 leads to It then follows that the equation does not have a solution and only which is equivalent to that is odd and is odd and,wehave It then follows from Lemma 21 that Since, the dimension of this code is equal to We shall need the following lemma later Lemma 55: Let is odd, is even Proof: First of all, we have It is also known that Since,wehave,wehave is odd is even (5) (6) (7)

4 DING AND HELLESETH: OPTIMAL TERNARY CYCLIC CODES FROM MONOMIALS 5901 It then follows that for all Hence for all Therefore,,let Then, It then follows from the discussions above that Theorem 56: Let Then, the ternary cyclic code has and only is odd, ie, and only is planar over Proof: We first prove the necessity of the condition Notice that We have We now prove that Condition C2 of Theorem 41 is satisfied To this end, we need to prove that there does not exist two distinct elements and in such that Suppose on the contrary that (8) has such a solution Then, adding 1 to both sides of both equations in (8) yields Define and It then follows from (9) that (8) (9) (10) Adding and to the first and second equation of (8) gives (11) It then follows that (12) is odd is even Suppose first that is even Then, divides Let be a generator of Define then Then, It is easily checked that Hence is a solution of This means that Condition C3 in Theorem 41 is not met So we have reached a contradiction This proves the necessity of the condition in this theorem We now prove the sufficiency of this condition is odd, it was proved in [11] that is planar It then follows from Theorem 52 that has This completes the proof of this theorem VI OPTIMAL CYCLIC CODES DEFINED BY APN MONOMIALS OVER In this section, we present seven classes of optimal cyclic codes defined by APN monomials over Theorem 61: The ternary cyclic code has is APN over Proof: Let be APN over Itthen follows from Lemma 51 that ByLemma 21, the dimension of this code is equal to Hence, the equation has three distinct solutions,and This is contrary to the assumption that is APN Note that hasatmosttwosolutions for any according to the definition of APN functions This completes the proof of Condition C2 in Theorem 41 Finally, we prove that Condition C3 is also satisfied Suppose on the contrary that the equation has a solution Then, ce and thus Dividing both sides of the equation with yields Therefore, is also a solution of the equation So this equation has three distinct solutions,and Thisis contrary to the assumption that is APN over Hence, Condition C3 in Theorem 41 is indeed met The desired conclusions of this theorem then follow from Theorem 41 The following is a summary of known APN monomials over Each of them gives a class of optimal ternary cyclic codes with 1), is odd 2), is odd 3), and is odd ([14]) 4), and is odd ([14]) 5), [28] 6) The exponent is defined by 7) The exponent is defined by

5 5902 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 59, NO 9, SEPTEMBER 2013 There are other APN monomials over for The reader is referred to [14] and [28] for details Example 62: Let and let Let be the generator of with Then, is a ternary cyclic code with and generator polynomial The dual of is a ternary cyclic code with This code is also optimal, while the optimal ternary cyclic code with the same in the database is not known to be cyclic VII OPTIMAL CYCLIC CODES DEFINED BY OTHER MONOMIALS OVER In the previous sections, optimal ternary cyclic codes from planar and APN monomials over were presented In this section, we construct several classes of optimal ternary cyclic codes with using monomials over that are neither planar nor APN We prove first the following theorem Theorem 71: Let The ternary cyclic code has andonly (a) is odd; (b) is even; (c) ;and (d) Proof: The code does not have a codeword of Hamming weight two and only is even Clearly, is even andonly is even Obviously, Conditions C2 and C3 in Theorem 41 are simultaneously satisfied and only the equation does not have a solution Let Notethat Conditions (a), (b), (13) and (14) together are equivalent to Conditions (a), (b), (c) and (d) together all the conditions of this theorem are met, we have It then follows from Lemma 21 that Note that Hence, for any Thus, The desired conclusions in this theorem then follow from Theorem 41 Example 72: Let and let Let be the generator of with Then, is a ternary cyclic code with and generator polynomial The dual of is a ternary cyclic code with This code has the same as the best known ternary linear code with in the database The upper bound on the minimum distance of any ternary linear code of length 242 and dimension 10 is 155 Lemma 73: Let is odd, is even (15) Proof: The proof is similar to that of Lemma 55 and is omitted Theorem 74: Let,andlet be even Then, the ternary cyclic code has The equation and only does not have a solution Proof: It is easily seen that The conclusions on the dimension then follow from Lemma 73 Since is even, the minimum distance We now prove that Tothis end, we prove that Conditions C2 or C3 in Theorem 41 are not satisfied Since is even, is even Define, where is a generator of Then, is even, we have (13) and It is easily seen that for any (14) In this case, Condition C3 in Theorem 41 is not met, and hence is odd, we have

6 DING AND HELLESETH: OPTIMAL TERNARY CYCLIC CODES FROM MONOMIALS 5903 In this case, Condition C3 in Theorem 41 is not met, and hence, The code of Theorem 74 is almost optimal when, and is optimal when and Open Problem 75: Let,andlet be odd Does the ternary cyclic code have? and confirmed by Magma Theorem 76: Let The ternary cyclic code has andonly (A) ;and (B) Proof: Let Conditions (A) and (B) be met We first prove that To this end, we prove that there is no such that divides Note that Hence, Clearly, Thus, the dimension of the code is equal to It is now time to prove that the minimum distance Suppose on the contrary that the equation has a solution Then, we have Multiplying both sides of this equation with yields whence It then follows from Condition (A) that This is contrary to the assumption that Therefore, Condition C2 in Theorem 41 is met Now suppose on the contrary that the equation has a solution Then,we have Multiplying both sides of this equation with gives whence and It then follows from Condition (B) that This is contrary to the assumption that Therefore, Condition C3 in Theorem 41 is satisfied Note that is even It then follows from Theorem 41 that Example 77: Let and let Let be the generator of with Then, is a ternary cyclic code with and generator polynomial The dual of is a ternary cyclic code with In the following, we list a number of open problems on the ternary cyclic codes Open Problem 78: Let Does the ternary cyclic code have and is prime? and confirmed by Magma Open Problem 79: Let Is it true that the ternary cyclic code has 1) is odd and ; 2) is odd; and 3)? For all and confirmed by Magma Open Problem 710: Let Let be odd and Isittruethattheternary cyclic code has is even? For all and confirmed by Magma Open Problem 711: Let Let be even What are the conditions on and under which the ternary cyclic code has? Open Problem 712: Let Let be even Is it true that the ternary cyclic code has one of the following conditions is met? 1), and 2), and For and confirmed by Magma Open Problem 713: Let Let and let be a prime Is it true that the ternary cyclic code has for all with? For, the answer to this question is positive and confirmed by Magma Open Problem 714: Let Let be an odd prime What are the conditions on under which the ternary cyclic code has? Open Problem 715: Let, where What are the conditions on and under which the ternary cyclic code has? VIII CONCLUDING REMARKS is planar or APN over, the code has andisthusoptimalhow-

7 5904 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 59, NO 9, SEPTEMBER 2013 ever, as demonstrated in this paper, the code may have the same when is neither planar nor APN It looks hard to completely characterize the ternary cyclic codes with In this paper, we presented nine open problems on the codes It would be nice some of these open problems could be settled Finally, we inform the reader that there are other examples of monomials that define optimal cyclic codes according to our Magma experimental data ACKNOWLEDGMENT The authors are grateful to the reviewers for their comments that improved the presentation and quality of this paper REFERENCES [1] T Beth and C Ding, On almost perfect nonlinear permutations, in Proc Adv Cryptol EUROCRYPT, NewYork,NY,USA, 1993, vol 765, Lecture Notes Comput Sci, pp [2] L Budaghyan and C Carlet, Classes of quadratic APN trinomials and hexanomials and related structures, IEEE Trans Inf Theory, vol54, no 5, pp , May 2008 [3] LBudaghyan,CCarlet, and G Leander, Two classes of quadratic APN binomials inequivalent to power functions, IEEE Trans Inf Theory, vol 54, no 9, pp , Sep 2008 [4] L Budaghyan, C Carlet, and G Leander, Constructing new APN functions from known ones, Finite Fields Appl, vol 15, no 2, pp , Apr 2009 [5] L Budaghyan, C Carlet, and A Pott, New classes of almost bent and almost perfect nonlinear functions, IEEE Trans Inf Theory, vol 52, no 3, pp , Mar 2006 [6] A Canteaut, P Charpin, and H Dobbertin, Weight divisibility of cyclic codes, highly nonlinear functions on, and crosscorrelation of maximum-length sequences, SIAMJDiscr Math,vol13,no1, pp , 2000 [7] C Carlet, P Charpin, and V Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des Codes Cryptogr, vol 15, pp , 1998 [8] C Carlet, C Ding, and J Yuan, Linear codes from highly nonlinear functions and their secret sharing schemes, IEEE Trans Inf Theory, vol 51, no 6, pp , Jun 2005 [9] P Charpin, Open problems on cyclic codes, in Handbook of Coding Theory, Part 1: Algebraic Coding,VSPless,WCHuffman,andR A Brualdi, Eds Amsterdam, The Netherlands: Elsevier, 1998, ch 11 [10] R T Chien, Cyclic decoding procedure for the Bose Chaudhuri Hocquenghem codes, IEEE Trans Inf Theory, vol10,no4, pp , Oct 1964 [11] R S Coulter and R W Matthews, Planar functions and planes of Lenz Barlotti class II, Des Codes Cryptogr, vol 10, pp , 1997 [12] P Dembowski and T G Ostrom, Planes oforder with collineation groups of order, Math Z, vol 193, pp , 1968 [13] C Ding and J Yuan, A family of skew Hadamard dference sets, J Combin Theory, ser A, vol 113, pp , 2006 [14] T Helleseth, C Rong, and D Sandberg, New families of almost perfect nonlinear power mappings, IEEE Trans Inf Theory, vol 45, no 2, pp , Mar 1999 [15] H Dobbertin, Almost perfect nonlinear power functions on : The Welch case, IEEE Trans Inf Theory, vol 45, no 4, pp , May 1999 [16] H Dobbertin, Almost perfect nonlinear power functions on : The Niho case, Inf Comput, vol151, pp 57 72, 1999 [17] G D Forney, On decoding BCH codes, IEEE Trans Inf Theory, vol 11, no 4, pp , Oct 1995 [18] R Gold, Maximal recursive sequences with 3-valued recursive crosscorrelation functions, IEEE Trans Inf Theory, vol 14, no 1, pp , Jan 1968 [19] T Kasami, The weight enumerators for several classes of subcodes of the second order binary Reed Muller codes, Inf Control, vol 18, pp , 1971 [20] H D L Hollmanna and Q Xiang, A proof of the Welch and Niho conjectures on cross-correlations of binary m-sequences, Finite Fields Appl, vol 7, no 2, pp , Apr 2001 [21] W C Huffman and V Pless, Fundamentals of Error-Correcting Codes Cambridge, UK: Cambridge Univ Press, 2003 [22] L Lidl and H Niederreiter, Finite Fields Cambridge, UK: Cambridge Univ Press, 1997 [23] J H van Lint and R M Wilson, On the minimum distance of cyclic codes, vol 32, no 1, pp 23 40, Jan 1986 [24] K Nyberg, Dferentially unorm mappings for cryptography, in Proc Adv Cryptol EUROCRYPT, New York, NY, USA, 1993, vol 765, Lecture Notes in Comput Sci, pp [25] E Prange, Some cyclic error-correcting codes with simple decoding algorithms Cambridge, MA, USA, 1958, Air Force Cambridge Research Center-TN [26] J Yuan, C Carlet, and C Ding, The weight distribution of a class of linear codes from perfect nonlinear functions, IEEE Trans Inf Theory, vol 52, no 2, pp , Feb 2006 [27] Z Zha, G M Kyureghyan, and X Wang, Perfect nonlinear binomials and their semields, Finite Fields Appl, vol 15, pp , 2009 [28] Z Zha and X Wang, Almost perfect nonlinear power functions in odd characteristic, IEEE Trans Inf Theory, vol 57, no 7, pp , Jul 2011 Cunsheng Ding (M 98 SM 05) was born in 1962 in Shaanxi, China He received the MSc degree in 1988 from the Northwestern Telecommunications Engineering Institute, Xian, China; and the PhD in 1997 from the University of Turku, Turku, Finland From 1988 to 1992 he was a Lecturer of Mathematics at Xidian University, China Before joining the Hong Kong University of Science and Technology in 2000, where he is currently Professor of Computer Science and Engineering, he was Assistant Professor of Computer Science at the National University of Singapore His research fields are cryptography and coding theory He has coauthored four research monographs, and served as a guest editor or editor for ten journals Dr Ding co-received the State Natural Science Award of China in 1989 Tor Helleseth (M 89 SM 96 F 97) received the Cand Real and Dr Philos degrees in mathematics from the University of Bergen, Bergen, Norway, in 1971 and 1979, respectively From , he was a Research Assistant at the Department of Mathematics, University of Bergen From , he was at the Chief Headquarters of Defense in Norway Since 1984, he has been a Professor in the Department of Informatics, University of Bergen During the academic years and , he was on sabbatical leave at the University of Southern Calornia, Los Angeles, and during , he was a Research Fellow at the Eindhoven University of Technology, Eindhoven, The Netherlands His research interests include coding theory and cryptology Prof Helleseth served as an Associate Editor for Coding Theory for the IEEE TRANSATIONS ON INFORMATION THEORY from 1991 to 1993 He was Program Chairman for Eurocrypt 1993 and for the Information Theory Workshop in 1997 in Longyearbyen, Norway He was a Program Co-Chairman for SETA04 in Seoul, Korea, and SETA06 in Beijing, China He was also a Program Co-Chairman for the IEEE Information Theory Workshop in Solstrand, Norway in 2007 During he served on the Board of Governors for the IEEE Information Theory Society In 1997 he was elected an IEEE Fellow for his contributions to coding theory and cryptography In 2004 he was elected a member of Det Norske Videnskaps-Akademi