On the p-ranks and Characteristic Polynomials of Cyclic Difference Sets

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1 Designs, Codes and Cryptography, 33, 23 37, 2004 # 2004 Kluwer Academic Publishers. Manufactured in The Netherlands. On the p-ranks and Characteristic Polynomials of Cyclic Difference Sets JONG-SEON NO School of Electrical Engineering and Computer Science, Seoul National University, Seoul , Korea DONG-JOON SHIN Division of Electrical and Computer Engineering, Hanyang University, Seoul , Korea TOR HELLESETH Department of Informatics, University of Bergen, N-5020 Bergen, Norway Communicated by: D. Jungnickel Received February 15, 2001; Revised September 24, 2002; Accepted October 9, 2002 Abstract. In this paper, the p-ranks and characteristic polynomials of cyclic difference sets are derived by expanding the trace expressions of their characteristic sequences. Using this method, it is shown that the 3-ranks and characteristic polynomials of the Helleseth Kumar Martinsen (HKM) difference set and the Lin difference set can be easily obtained. Also, the p-rank of a Singer difference set is reviewed and the characteristic polynomial is calculated using our approach. Keywords: cyclic difference set, characteristic polynomial, p-rank AMS Classification: 05B10, 11B50 1. Introduction It is well-known that a cyclic difference set with Singer parameters ð2 n 1; 2 n 1 1; 2 n 2 1Þ [7,17] is equivalent to a binary sequences of period 2 n 1 with ideal autocorrelation [4,5,14 16]. Recently, nonbinary sequences with ideal autocorrelation have been investigated. Helleseth et al. [8] found ternary sequences with ideal autocorrelation that turned out to be cyclic difference sets with Singer parameters ðð3 3k 1Þ=ð3 1Þ; ð3 3k 1 1Þ=ð3 1Þ; ð3 3k 2 1Þ=ð3 1ÞÞ [9,13]. Lin [12] found a new family of ternary sequences of period 3 n 1 and conjectured that it has the ideal autocorrelation property, when n ¼ 2m þ 1. Under this assumption, a cyclic difference set (Lin difference set) with Singer parameters ðð3 n 1Þ= ð3 1Þ; ð3 n 1 1Þ=ð3 1Þ; ð3 n 2 1Þ=ð3 1ÞÞ can be obtained [9,13]. In this paper, the p-ranks and characteristic polynomials of cyclic difference sets are derived by expanding the trace expressions of their characteristic sequences. Using this method, it is shown that the 3-ranks and characteristic polynomials of the Helleseth Kumar Martinsen (HKM) difference set and the Lin difference set can be

2 24 NO ET AL. easily obtained. Also, the p-rank of a Singer difference set is reviewed and its characteristic polynomial is calculated using our approach. 2. Preliminaries Let D be a ðv; k; lþ difference set [3,10] defined as a set of k distinct residues modulo v expressed by D ¼fc 1 ; c 2 ; c 3 ;...; c k g: ð1þ Then each non-zero residue occurs exactly l times among the kðk 1Þ differences c i c j ; i 6¼ j and thus it satisfies lðv 1Þ ¼kðk 1Þ: The complementary difference set of D is a ðv; v k; v 2k þ lþ difference set defined as D ¼ Z v nd; where Z v is the ring of integers modulo v. Then for integers a and b, a set ad þ b is defined as ad þ b ¼fa? c 1 þ b; a? c 2 þ b; a? c 3 þ b;...; a? c k þ bg; where a? c i þ b is taken modulo v. For integers a and b, two ðv; k; lþ difference sets D 1 and D 2 are said to be inequivalent if D 1 is distinct from ad 2 þ b for any integer a and b, 1 a v 1, 0 b v 1, where a is relatively prime to v. The characteristic sequence of a cyclic difference set D in (1) is defined as the binary sequence 1; if t [ D; sðtþ ¼ 0; if t 6[ D; and the characteristic sequence of its complementary difference set D is defined as 1; if t [ D; s c ðtþ ¼ 0; if t 6[ D; ¼ 1 sðtþ: The characteristic polynomial of a cyclic difference set D is defined as a leastdegree linear recursion equation over F p of the characteristic sequence sðtþ of D. The p-rank [6] of a cyclic difference set D is defined as the degree of characteristic polynomial of the cyclic difference set D. In order to prove the inequivalence of two cyclic difference sets, the p-rank is often used. However, it is very difficult to find the p-ranks of cyclic difference sets.

3 ON THE p-ranks AND CHARACTERISTIC POLYNOMIALS 25 Let n ¼ e? m > 1 for some positive integers e and m. Then the trace function tr n m ð? Þ is the mapping from the finite field F p n to its subfield F pm defined by Lidl and Niederreiter [11] tr n mðxþ ¼e 1 x? pm i ; where x is an element in F p n. It is easy to check that the trace function satisfies the following: i. tr n m ðax þ byþ ¼atrn m ðxþþbtrn m ð yþ, for all a; b [ F p m; x; y [ F p n. ii. tr n m ðxpm Þ¼tr n m ðxþ, for all x [ F p n. iii. tr n 1 ðxþ ¼trm 1 ðtrn m ðxþþ, for all x [ F p n. Using the trace function, a p-ary m-sequence mðtþ of period p n 1 can be expressed as mðtþ ¼tr n 1 ðat Þ; ð2þ where p is a prime and a is a primitive element of F p n. 3. p-ranks and Characteristic Polynomials of Cyclic Difference Sets As will be given in the following theorem and lemma, the characteristic sequences of a cyclic difference set and its complementary difference set can be expressed by using the trace expressions and the p-rank can be calculated by counting the number of terms in the trace expression. Moreover, by finding the minimal polynomials corresponding to this trace expression, we can easily obtain the characteristic polynomial. This method will be used to find the 3-ranks and characteristic polynomials of the HKM and the Lin difference sets in the following sections. Let f ðtþ be a function from the finite field F p n to its subfield F p m. Then we have the following relation: ½ f ðtþš pm 1 ¼ 1; if f ðtþ 6¼ 0; 0; if f ðtþ ¼0: Therefore, using this relation, the characteristic sequence of a cyclic difference set is given in the following theorem. THEOREM 1. Assume that a ðv; k; lþ cyclic difference set D and its complementary difference set D are defined as D ¼ft j f ðtþ ¼0; 0 t < vg; D ¼ft j f ðtþ 6¼ 0; 0 t < vg; ð3þ ð4þ

4 26 NO ET AL. where f(t) is given as a summation of the trace function from the finite field F p n to its subfield F p m, that is f ðtþ ¼ b a? tr n m ðaat Þ; a [ I for some index set I; b a [ F p m and a primitive elements a in F p n. Then the characteristic sequences of the cyclic difference set D and its complementary difference set D can be expressed as and sðtþ ¼1 s c ðtþ; 0 t < v; s c ðtþ ¼½f ðtþš pm 1 ; 0 t < v: ð5þ In order to find the p-rank of the cyclic difference set D, we have to expand the trace expression in (5) as given in the following lemma. LEMMA 2. Let D and Dbeaðv; k; lþ cyclic difference set and its complementary difference set defined by (3) and (4), respectively. Suppose that the trace expression of the characteristic sequence s c ðtþ of D can be expanded as s c ðtþ ¼½fðtÞŠ pm 1 " # p m 1 ¼ a [ I b a? tr n m ðaat Þ ¼ c j a jt ; j [ J ð6þ where J is an index set and c j [ Fp. Then the p-rank of the complementary difference set D is given as jjj, which means that the degree of the characteristic polynomial of Dis jjj. Further, the p-rank of the cyclic difference set D is given as jjjþ1, if jdj ¼v kis divisible by p. Proof. Let gðxþ and g c ðxþ be the characteristic polynomials over F p of a cyclic difference set D and its complementary difference set D, respectively. Since there is the relationship between s c ðtþ and sðtþ given as sðtþ ¼1 s c ðtþ; their characteristic polynomials are related as aðxþ gðxþ ¼ 1 bðxþ x 1 g c ðxþ ¼ g cðxþ ðx 1ÞbðxÞ ; ðx 1Þ? g c ðxþ

5 ON THE p-ranks AND CHARACTERISTIC POLYNOMIALS 27 where the polynomials aðxþ and bðxþ are relatively prime to gðxþ and g c ðxþ, respectively. If jdj ¼v k is divisible by p, then the number of 1 s in one period of the characteristic sequence of the cyclic difference set D is a multiple of p. Therefore, the polynomial expression of the characteristic sequence possesses the factor x 1 and this cancels the factor x 1 in the characteristic polynomial, which means that the characteristic polynomial g c ðxþ is not divisible by x 1. Therefore, the polynomial g c ðxþ ðx 1ÞbðxÞ is relatively prime to ðx 1Þg c ðxþ and we obtain gðxþ ¼ðx 1Þg c ðxþ; ð7þ if jdj ¼v k is divisible by p. Therefore, the p-rank of the cyclic difference set D is jjjþ1, if jdj ¼v k is divisible by p. & If (6) is expressed as a summation of trace functions as ½ f ðtþš pm 1 ¼ k j n;k>1 c ak? tr k 1 ðaakt k Þ; a k [ J k where c ak [ Fp ; J k s are index sets and a k is a primitive element of F p k, the characteristic polynomials of the cyclic difference set D and its complementary difference set D can be expressed as in the following theorem. THEOREM 3. If the characteristic sequence of a cyclic difference set D is expressed as s c ðtþ ¼ c ak? tr k 1 ðaakt k Þ; a k [ J k kjn;k>1 where c ak [ Fp and if jdj ¼v l is divisible by p, then the characteristic polynomials gðxþ and g c ðxþ of the cyclic difference set D and its complementary difference set D are given as follows: gðxþ ¼ðx 1Þ? g c ðxþ; g c ðxþ ¼ Y Y M ak ðxþ; a k [ J k kjn;k>1 ð8þ ð9þ where M ak ðxþ is the minimal polynomial of the element a a k k [ F p k. Note that the p-rank of D is one bigger than that of D. For the cyclic difference set with Singer parameters ððq n 1Þ=ðq 1Þ; ðq n 1 1Þ=ðq 1Þ; ðq n 2 1Þ=ðq 1ÞÞ, where q ¼ p m ; v k ¼ q n 1 ¼ p mðn 1Þ is divisible by p. Therefore, the above theorem can be applied to find the p-ranks and characteristic polynomials of the cyclic difference set with Singer parameters and its complementary difference set.

6 28 NO ET AL Ranks and Characteristic Polynomials of HKM Difference Sets Recently, Helleseth et al. [8] introduced a new ternary sequence ð p ¼ 3Þ with ideal autocorrelation. It turned out to be a cyclic difference set (HKM difference set) with Singer parameters [9,13] and the p-ranks of the HKM difference set and its complementary difference set are derived in the following theorem. THEOREM 4. The HKM difference set with parameters ðð3 n 1Þ=ð3 1Þ; ð3 n 1 1Þ= ð3 1Þ; ð3 n 2 1Þ=ð3 1ÞÞ is defined as D ¼ tjtr n 1 ðat Þþtr n 1 ðadt Þ¼0; 0 t < 3n 1 ; ð10þ 3 1 where n ¼ 3k > 3, d ¼ 3 2k 3 k þ 1, and a is a primitive element of F 3 n. Then the 3-ranks of the HKM difference set D and its complementary difference set D are given as 2n 2 2n þ 1 and 2n 2 2n, respectively. Proof. Let x ¼ a t. Then we can expand the square of the trace function as tr n 2 1 ðat Þ ¼ tr n 2 1 ðxþ ¼ tr n 1 ðxþ? trn 1 ðxþ ¼ tr n 1 ðx? trn 1 ðxþþ ¼ tr n 1 ðx? ðx þ x3 þ x 32 þþx 3n 1 ÞÞ ¼ n 1 tr n 1 ðx1þ3i Þ ¼ tr n 1 ðx1þ1 Þþ n 1 tr n 1 Þ ðx1þ3i 8 >< tr n 1 ðx1þ1 Þþ2? ¼ >: ðn 1Þ=2 P tr n 1 ðx1þ3i Þ; tr n 1 ðx1þ1 Þþ2? tr n=2 1 ðx 1þ3ðn=2Þ Þþ2? ðn=2þ 1 P for n ¼ odd; tr n 1 ðx1þ3i Þ; for n ¼ even: The characteristic sequence of the cyclic difference set D is given as tr n 2: 1 ðxþþtrn 1 ðxd Þ ð11þ Using the expansion of the square of trace function, we can also expand (11) as follows.

7 ON THE p-ranks AND CHARACTERISTIC POLYNOMIALS 29 (i) n ¼ 2m þ 1 (odd case) tr n 2 1 ðxþþtrn 1 ðxd Þ ¼ tr n 2 1 ðxþ þ tr n 1 ðx d 2 Þ þ 2? tr n 1 ðxþ? tr n 1 ðxd Þ Since we have Using we obtain Therefore, ¼ tr n 1 ðx1þ1 Þþ2? þ 2? ðn 1Þ=2 ¼ tr n 1 ðx1þ1 Þþ2? þ 2? ðn 1Þ=2 ðn 1Þ=2 tr n 1 ðx1þ3i Þþtr n 1 ðxð1þ1þd Þ tr n 1 ðxð1þ3i Þd Þþ2? tr n 1 ðxd? tr n 1 ðxþþ ðn 1Þ=2 tr n 1 ðx1þ3i Þþtr n 1 ðxð1þ1þd Þ tr n 1 Þd ðxð1þ3i Þþ2? n 1 tr n 1 ðxdþ3i Þ: d? ð3 k þ 1Þ ¼3 3k þ 1 ¼ 3 3k 1 þ 2 : 2 modð3 3k 1Þ; tr n 1 ðxð1þ3k Þd Þ¼tr n 1 ðx1þ1 Þ: d þ 3 k ¼ 3 2k þ 1; tr n 1 ðxdþ3k Þ¼tr n 1 ðx32k þ1 Þ¼tr n 1 ðx3k þ1 Þ: tr n 2 m 1 ðxþþtrn 1 ðxd Þ ¼ 2? þ 2? ; i6¼k n 1 ; i6¼k tr n 1 ðx1þ3i Þþtr n 1 ðxð1þ1þd Þþ2? tr n 1 ðxdþ3i Þþtr n 1 ðx1þ3k Þ: m ;i6¼k tr n 1 ðxð1þ3i Þd Þ It is easy to prove that for n > 3, all exponents belong to different cyclotomic coset of size n. Therefore, the 3-rank of D is ðm 1Þn þ n þðm 1Þn þðn 1Þn þ n ¼ 2? nðn 1Þ; and the 3-rank of the HKM difference set D is therefore 2n 2 2n þ 1. ð12þ

8 30 NO ET AL. (ii) n ¼ 2m (even case): Similar to the odd case, we expand tr n 2 1 ðxþþtrn 1 ðxd Þ ¼ tr n 2 1 ðxþ þ tr n 1 ðx d 2 Þ þ 2? tr n 1 ðxþ? tr n 1 ðxd Þ ¼ tr n 1 ðx1þ1 Þþ2? tr m 1 ðx1þ3m Þþ2? m 1 tr n 1 ðx1þ3i Þ Using we have and since we get þ tr n 1 ðxð1þ1þd Þþ2? tr m 1 Þd ðxð1þ3m Þþ2? m 1 tr n 1 Þd ðxð1þ3i Þ þ 2? n 1 tr n 1 ðxdþ3i Þ: ð1 þ 3 k Þ? d ¼ 2 mod ð3 3k 1Þ; tr n 1 ðxð1þ3k Þd Þ¼tr n 1 ðx2 Þ; d þ 3 k ¼ 3 2k þ 1; tr n 1 ðxdþ3k Þ¼tr n 1 ðx1þ32k Þ¼tr n 1 ðx1þ3k Þ: Adding up the same trace terms, we have tr n 2 1 ðxþþtrn 1 ðxd Þ ¼ 2? tr m 1 ðx ð1þ3mþ Þþ2? m 1 ;i6¼k tr n 1 ðx1þ3i Þ þ tr n 1 ðxð1þ1þd Þþ2? tr m 1 ðxð1þ3m Þd Þþ2? þ 2? n 1 ;i6¼k tr n 1 ðxdþ3i Þþtr n 1 ðx1þ3k Þ: m 1 ;i6¼k tr n 1 ðxdð1þ3iþ Þ It can be shown that all exponents belong to different cosets of size n or m. Therefore the 3-rank of the difference set D is m þðm 2Þn þ n þ m þðm 2Þn þðn 1Þn þ n ¼ 2? nðn 1Þ; and from (8), the 3-rank of the HKM difference set D is therefore 2n 2 2n þ 1. & ð13þ

9 ON THE p-ranks AND CHARACTERISTIC POLYNOMIALS 31 Note: Using (12), for n ¼ 3, it can be easily derived that the 3-ranks of the HKM difference set D and its complementary difference set D are 7 and 6, respectively. From (12) and (13), we can derive the characteristic polynomials of the HKM difference set and its complementary difference set as in the following theorem. THEOREM 5. Let D be the HKM difference set with parameters ðð3 n 1Þ=ð3 1Þ; ð3 n 1 1Þ=ð3 1Þ; ð3 n 2 1Þ=ð3 1ÞÞ defined in (10). Then the characteristic polynomials of the HKM difference set D and its complementary difference set D are: For n ¼ 2m þ 1: gðxþ ¼ðx 1Þg c ðxþ; g c ðxþ ¼M 2d ðxþm 1þ3 kðxþ Ym 6 Ym ;i6¼k ;i6¼k M ð1þ3i ÞdðxÞ Yn 1 M 1þ3 iðxþ ;i6¼k M dþ3 iðxþ: For n ¼ 2m: gðxþ ¼ðx 1Þg c ðxþ; g c ðxþ ¼M 2d ðxþm 1þ3 mðxþm ð1þ3m ÞdðxÞM 1þ3 kðxþ Ym 1 6 Ym 1 ;i6¼k M ð1þ3 i ÞdðxÞ Y n 1 ; i6¼k M dþ3 iðxþ: ;i6¼k M 1þ3 iðxþ Note: For n ¼ 3, the characteristic polynomials of the HKM difference set D and its complementary difference set D are derived as ðx 1ÞM 4 ðxþm 8 ðxþ and M 4 ðxþm 8 ðxþ, respectively Ranks and Characteristic Polynomials of Lin Difference Sets Lin [12] has conjectured that the family of ternary sequences cðtþ ¼tr n 1 ðat Þþtr n 1 ðadt Þ has the ideal autocorrelation property, where n ¼ 2m þ 1andd ¼ 2? 3 m þ 1. If this is true, then it gives a cyclic difference set (Lin difference set) with Singer parameters [9,13]. Under this assumption, we can derive the 3-ranks and characteristic polynomials of the Lin difference set and its complementary difference set as in the following theorems.

10 32 NO ET AL. THEOREM 6. The Lin difference set with parameters ðð3 n 1Þ=ð3 1Þ; ð3 n 1 1Þ= ð3 1Þ; ð3 n 2 1Þ=ð3 1ÞÞ is defined as D ¼ tjtr n 1 ðat Þþtr n 1 ðadt Þ¼0; 0 t < 3n 1 ; ð14þ 3 1 where n ¼ 2m þ 1 > 3, d ¼ 2? 3 m þ 1 and a is a primitive element of F 3 n. Then the 3-ranks of the Lin difference set D and its complementary difference set D are given as 2n 2 2n þ 1 and 2n 2 2n, respectively. Proof. Let x ¼ a t. Then the characteristic sequence of the difference set D is tr n 2: 1 ðxþþtrn 1 ðxd Þ ð15þ Using the expansion of the square of the trace function in the proof of the previous theorem, we can expand (15) as follows. tr n 1 ðat Þþtr n 2 1 ðadt Þ ¼½tr n 1 ðxþš 2 þ tr n 2 1 ðxd Þ þ 2? tr n 1 ðxþ? tr n 1 ðxd Þ ¼ tr n 1 ðx1þ1 Þþ2? m tr n 1 Þþtr n ðx1þ3i 1 ðxð1þ1þd Þ Since we have and using we obtain þ 2? m d þ 3 m ¼ 2? 3 m þ 1 þ 3 m ¼ 3 mþ1 þ 1; tr n 1 ðxdþ3m Þ¼tr n 1 ðx1þ3m Þ; d þ 3 n 1 ¼ 3 n 1 þ 2? 3 m þ 1; tr n 1 Þd Þþ2? n 1 tr n ðxð1þ3i 1 Þ: ðxdþ3i tr n 1 ðxdþ3n 1 Þ¼tr n? 1 ðx2 3mþ1þ3þ1 Þ ¼ tr n? 1 ðx2 ð3mþ1þ2þ Þ ¼ tr n? 1 ðx2 ð1þ2? 3mÞ Þ ¼ tr n 1 ðxð1þ1þd Þ:

11 ON THE p-ranks AND CHARACTERISTIC POLYNOMIALS 33 Adding up the same trace terms, we have tr n 1 ðxþþtrn 1 ðxd Þ 2 ¼ tr n 1 ðx 1þ1 Þþ2? m 1 tr n 1 ðx1þ3i Þþtr n 1 ðx1þ3m Þ þ 2? m tr n 1 ðxð1þ3i Þd Þþ2? n 2 ;i6¼m tr n 1 ðx3i þd Þ: ð16þ It can be shown that for n > 3, all exponents belong to different cosets of size n. Therefore, the 3-rank of the difference set D is n þðm 1Þn þ n þ m? n þ nðn 2Þ ¼2nðn 1Þ; and from (8), the 3-rank of the Lin difference set D is 2n 2 2n þ 1. & Note: For n ¼ 3, the Lin difference set is identical to the HKM difference set. Therefore, the 3-ranks and characteristic polynomials of the Lin difference set D and its complementary difference set D are the same as those of the HKM difference set and its complementary difference set. From (16), we can derive the characteristic polynomials of the Lin difference set and its complementary difference set as in the following theorem. THEOREM 7. Let D be the Lin difference set with parameters ðð3 n 1Þ=ð3 1Þ; ð3 n 1 1Þ=ð3 1Þ; ð3 n 2 1Þ=ð3 1ÞÞ defined in (14). Then the characteristic polynomials of the Lin difference set D and its complementary difference set D are given as: gðxþ ¼ðx 1Þg c ðxþ; g c ðxþ ¼M 2 ðxþm 1þ3 mðxþ Ym 1 M 1þ3 iðxþ 6 Ym M ð1þ3i ÞdðxÞ Y n 2 ;i6¼m M dþ3 iðxþ: 6. Characteristic Polynomials of Singer Difference Sets The p-rank of a Singer difference set is well-known [1,2,6,18]. In this section, the p-ranks of a Singer difference set and its complementary difference set are reviewed as given in the following theorem. Also the characteristic polynomials are obtained by using our approach.

12 34 NO ET AL. THEOREM 8 [1,2,6,18]. The Singer difference set with parameters ððq n 1Þ=ðq 1Þ; ðq n 1 1Þ=ðq 1Þ; ðq n 2 1Þ=ðq 1ÞÞ is defined as D ¼ tjtr ns s ðat Þ¼0; 0 t < qn 1 ; q 1 where q ¼ p s and a is a primitive element of F q n. Then the p-ranks of the Singer difference set D and its complementary difference set D are given as s p þ n 2 þ 1 and n 1 s p þ n 2 ; n 1 respectively. The sketchy proof of Theorem 8 is given here and it will be used to derive the characteristic polynomials in the following theorem. Let x ¼ a t. Then the characteristic sequence of the difference set D is given as tr ns s ðxþ p s 1. By using p s 1 ¼ðp 1Þp s 1 þðp 1Þp s 2 þþðp 1Þp þðp 1Þ, we get where s ðxþ h p s 1 ¼ ½tr ns s ðxþšp 1 tr ns ¼ ðl i;0 ;l i;1 ;...;l i;n 1 Þ [ I 0 i p s 1 þp s 2 þþpþ1 ðl i;0 ;l i;1 ;...;l i;n 1 Þ [ I s 1 Y s 1 p 1 l i;0 ; l i;1 ;...; l i;n 1 x cðlþ ; cðlþ ¼ n 1 j¼0 s 1 js p l i; j p i ; ( ) I i ¼ ðl i;0 ; l i;1 ;...; l i;n 1 Þj n 1 l i; j ¼ p 1; l i; j 0 ; 0 i s 1: j¼0 ð17þ First, it will be shown that all cðlþ s are distinct mod q n 1. Suppose cðlþ ¼cðl 0 Þ mod q n 1. The value of cðlþ is upper-bounded by q n 1, where the equality holds if and only if l i; j ¼ p 1 for all i; j. Since this equality cannot happen, cðlþ < q n 1 and the mod q n 1 can be ignored. By comparing the coefficients l i; j and l 0 i; j of piþjs in the expansion of cðlþ and cðl 0 Þ, it is easy to show that l i; j ¼ l 0 i; j and therefore, all cðlþ s are distinct for different l. There are p þ n 2 n 1 cases satisfying P n 1 l j;i ¼ p 1 and the corresponding coefficient values are in the form

13 ON THE p-ranks AND CHARACTERISTIC POLYNOMIALS 35 Y s 1 p 1 6¼ 0 mod p: l i;0 ; l i;1 ;...; l i;n 1 Therefore, the p-ranks of Singer difference set and its complementary difference set are given as s p þ n 2 þ 1 and n 1 s p þ n 2 ; n 1 respectively. From (17), we can derive the characteristic polynomials of a Singer difference set and its complementary difference set as in the following theorem. THEOREM 9. The Singer difference set with parameters ððq n 1Þ=ðq 1Þ; ðq n 1 1Þ= ðq 1Þ; ðq n 2 1Þ=ðq 1ÞÞ is defined as D ¼ tjtr ns s ðat Þ¼0; 0 t < qn 1 ; q 1 where q ¼ p s and a is a primitive element of F q n. Then the characteristic polynomials of Singer difference set D and its complementary difference set D are given as: gðxþ ¼ðx 1Þg c ðxþ; g c ðxþ ¼ Y M cðlþ ðxþ; cðlþ [ I where P I is a set of all coset leaders of the cosets including the elements s 1 P n 1 j¼0 l i; jp iþjs satisfying P n 1 j¼0 l i; j ¼ p 1; 0 i < s. Proof. From the proof of the previous theorem, we get cðlþ ¼p 0 ðl 0;0 þ l 1;0 p 1 þþl s 1;0 p s 1 Þ þ p s ðl 0;1 þ l 1;1 p 1 þþl s 1;1 p s 1 Þ.. þ p ðn 1Þs ðl 0;n 1 þ l 1;n 1 p 1 þþl s 1;n 1 p s 1 Þ; where P n 1 l j;i ¼ p 1 for all j. It is easy to show that cðlþ? p also appears as one of the valid cðl Þ s and has the same coefficient value as that of cðl Þ case. Hence, the theorem follows. & EAMPLE 10. Consider a Singer difference set with parameters ðð3 6 1Þ=ð3 2 1Þ; ð3 4 1Þ=ð3 2 1Þ; ð3 2 1Þ=ð3 2 1ÞÞ ¼ ð91; 10; 1Þ which is defined by D ¼ tjtr 6 2 ðat Þ¼0; 0 t < ¼ 91 ;

14 36 NO ET AL. where p ¼ 3 and a is a primitive element of F 3 6. Using the primitive polynomial x 6 þ 2x þ 2 ¼ 0, it can be shown that f0; 7; 19; 21; 57; 58; 63; 67; 80; 83g forms this Singer difference set. We can obtain seven cosets from cðlþ ¼l 0;0 þ l 1;0 3 þ l 0;1 3 2 þ l 1;1 3 3 þ l 0;2 3 4 þ l 1;2 3 5 where P 2 j¼0 l i; j ¼ 2; 0 i < 2. The valid values for ðl i;0 ; l i;1 ; l i;2 Þ; 0 i < 2, are among fð2; 0; 0Þ; ð0; 2; 0Þ; ð0; 0; 2Þ; ð1; 1; 0Þ; ð1; 0; 1Þ; ð0; 1; 1Þg. Therefore, the number of possible nonzero exponents cðl Þ for i ¼ 0; 1 is 666 ¼ 36 and it is the same value as 2 3 þ 3 2 : 3 1 We can select one value (coset leader) from each coset and the following values 0 1 l 0;0 l l 0;1 l 1;1 A; l 0;2 l 1;2 form seven such coset leaders < A : ; ; and the corresponding cðl Þ values are ð8; 16; 32; 40; 56; 64; 112Þ. Therefore, the characteristic polynomials of a Singer difference set with parameters ð91; 10; 1Þ and its complementary difference set are given as: where gðxþ ¼ðx 1Þg c ðxþ; g c ðxþ ¼M 8 ðxþm 16 ðxþm 32 ðxþm 40 ðxþm 56 ðxþm 64 ðxþm 112 ðxþ; M 8 ðxþ ¼x 6 þ x 4 þ x 3 þ 1; M 16 ðxþ ¼x 6 þ 2x 5 þ x 4 þ x 3 þ 2x 2 þ 1; M 32 ðxþ ¼x 6 þ x 5 þ x 4 þ 2x 3 þ x þ 1; M 40 ðxþ ¼x 6 þ x 5 þ 2x 4 þ 2x 3 þ 1; M 56 ðxþ ¼x 3 þ 2x 2 þ 2x þ 2; M 64 ðxþ ¼x 6 þ x 5 þ 2x 3 þ x 2 þ 2x þ 1; M 112 ðxþ ¼x 3 þ 2x þ 2:

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