On the Riemann hypothesis for self-dual weight enumerators of genus three
aa r X i v : . [ m a t h . N T ] N ov On the Riemann hypothesis for self-dual weightenumerators of genus three
Koji Chinen ∗ and Yuki Imamura ∗∗ Abstract
In this note, we give an equivalent condition for a self-dual weight enumerator of genusthree to satisfy the Riemann hypothesis. We also observe the truth and falsehood of theRiemann hypothesis for some families of invariant polynomials.
Key Words:
Zeta function for codes; Invariant polynomial ring; Riemann hypothesis.
Mathematics Subject Classification:
Primary 11T71; Secondary 13A50, 12D10.
Zeta functions for linear codes were introduced by Iwan Duursma [6] in 1999 and they haveattracted attention of many mathematicians:
Definition 1.1
Let C be an [ n, k, d ] -code over F q ( q = p r , p is a prime) with the Hammingweight enumerator W C ( x, y ) . Then there exists a unique polynomial P ( T ) ∈ R [ T ] of degree atmost n − d such that P ( T )(1 − T )(1 − qT ) ( y (1 − T ) + xT ) n = · · · + W C ( x, y ) − x n q − T n − d + · · · . (1.1) We call P ( T ) and Z ( T ) = P ( T ) / (1 − T )(1 − qT ) the zeta polynomial and the zeta function of W ( x, y ) , respectively. If C is self-dual, then P ( T ) satisfies the functional equation (see [7, § Theorem 1.2 If C is self-dual, then we have P ( T ) = P (cid:16) qT (cid:17) q g T g , (1.2) where g = n/ − d . ∗ Department of Mathematics, School of Science and Engineering, Kindai University. 3-4-1, Kowakae, Higashi-Osaka, 577-8502 Japan. E-mail: [email protected] ∗∗ interprism Inc. Kurihara BLD 2F Nihonbashi-kakigara-cho 2-12-8 Chuo-ku Tokyo, 103-0014 Japan. E-mail:[email protected] g is called the genus of C . It is appropriate to formulate the Riemann hypothesisas follows: Definition 1.3
The code C satisfies the Riemann hypothesis if all the zeros of P ( T ) have thesame absolute value / √ q . The reader is referred to [8] and [9] for other results by Duursma.
Remark.
The definition of the zeta function can be extended to much wider classes of invariantpolynomials: let W ( x, y ) be a polynomial of the form W ( x, y ) = x n + n X i = d A i x n − i y i ∈ C [ x, y ] ( A d = 0) (1.3)which satisfy W σ q ( x, y ) = ± W ( x, y ) for some q ∈ R , q > q = 1, where σ q = 1 √ q (cid:18) q − − (cid:19) (the MacWilliams transform) (1.4)and the action of σ = (cid:18) a bc d (cid:19) on a polynomial f ( x, y ) ∈ C [ x, y ] is defined by f σ ( x, y ) = f ( ax + by, cx + dy ). Then we can formulate the zeta function and the Riemann hypothesis for W ( x, y ) in the same way as Definitions 1.1 and 1.3. For the results in this direction, the readeris referred to [1]–[5], for example. We should also note that we must assume d, d ⊥ ≥
2, where d ⊥ is defined by W σ q ( x, y ) = B x n + B d ⊥ x n − d ⊥ y d ⊥ + · · · , when considering the zeta function of W ( x, y ).We do not know much about the Riemann hypothesis for self-dual weight enumerators,but one of the remarkable results is the following theorem by Nishimura [11, Theorem 1], anequivalent condition for a self-dual weight enumerator of genus one to satisfy the Riemannhypothesis: Theorem 1.4 (Nishimura)
A self-dual weight enumerator W ( x, y ) = x d + A d x d y d + · · · sat-isfies the Riemann hypothesis if and only if √ q − √ q + 1 (cid:18) dd (cid:19) ≤ A d ≤ √ q + 1 √ q − (cid:18) dd (cid:19) . (1.5)Nishimura also deduces the following, the case of genus two ([11, Theorem 2]): Theorem 1.5 (Nishimura)
A self-dual weight enumerator W ( x, y ) = x d +2 + A d x d +2 y d + · · · satisfies the Riemann hypothesis if and only if the both roots of the quadratic polynomial A d X − (cid:18) ( d − q ) A d + d + 1 d + 2 A d +1 (cid:19) X − ( d + 1)( q + 1) (cid:18) A d + A d +1 d + 2 (cid:19) + ( q − (cid:18) d + 2 d (cid:19) (1.6) are contained in [ − √ q, √ q ] . The purpose of this article is to establish an analogous equivalent condition for the case of genusthree. Our main result is the following: 2 heorem 1.6
A self-dual weight enumerator W ( x, y ) = x d +4 + A d x d +4 y d + · · · satisfies theRiemann hypothesis if and only if all the roots of the cubic polynomial f X + f X + f X + f (1.7) are contained in [ − √ q, √ q ] , where f i is defined as follows. f = A d ,f = ( q − d ) A d − d + 1 d + 4 A d +1 ,f = 12 ( d − qd + d − q ) A d + ( d − q + 1) d + 1 d + 4 A d +1 + ( d + 1)( d + 2)( d + 3)( d + 4) A d +2 ,f = 12 ( q + 1)( d + 3 d − q + 2) A d + ( q + 1)( d + 1)( d + 2) A d +1 d + 4+ ( q + 1) ( d + 1)( d + 2)( d + 3)( d + 4) A d +2 − ( q − (cid:18) d + 4 d + 4 (cid:19) . By this theorem, we can verify the truth of the Riemann hypothesis of W ( x, y ) only by threeparameters A d , A d +1 , A d +2 (the number of parameters which are needed coincides with the genus g , see [11]). Moreover, in many cases, we have A d +1 = 0 and the verification of the Riemannhypothesis is simplified.Theorem 1.6 leads us to the consideration of the truth or falsehood of the Riemann hypothesisas the numbers q and n vary. As was mentioned in Remark before, q can take other numbers thanprime powers. In this context, we can notice the tendency that the Riemann hypothesis becomesharder to hold if n or q are larger. Some of the results in [3] and [4] also support it. Theorem1.6 can illustrate this tendency by considering a certain sequence of invariant polynomials, thatis W n,q ( x, y ) = ( x + ( q − y ) n . (1.8)In Section 2, we give a proof of Theorem 1.6. In Section 3, we observe the behavior of W n,q ( x, y ), give some theoretical and experimental results, and state a conjecture on their Rie-mann hypothesis. Let W ( x, y ) = x d +4 + P d +4 i = d A i x d +4 − i y i be a self-dual weight enumerator. Using the functionalequation (1.2) (note that g = 3 in our case), we can assume that the zeta polynomial P ( T ) of W ( x, y ) is of the form P ( T ) = a + a T + a T + a T + a qT + a q T + a q T . We obtain another expression of P ( T ) because 1 /qα is a root of P ( T ) if P ( α ) = 0: P ( T ) = a q Y i =1 ( T + b i T + 1 /q ) . (2.1)3omparing the coefficients, we get b + b + b = a /a q,b b + b b + b b = ( a − a q ) /a q ,b b b = ( a − a q ) /a q . Thus b i is the roots of the following cubic polynomial: a q X − a q X + ( a − a q ) qX − a + 2 a q. (2.2)Considering the distribution of the roots of each factor T + b i T + 1 /q in (2.1), we can see thata self-dual weight enumerator W ( x, y ) of genus three satisfies the Riemann hypothesis if andonly if b , b and b are contained in [ − / √ q, / √ q ]. By change of variable in (2.2), we get thefollowing: Lemma 2.1 W ( x, y ) satisfies the Riemann hypothesis if and only if all the roots of the polyno-mial a X − a X + ( a − a q ) X − a + 2 a q (2.3) are contained in [ − √ q, √ q ] . Our next task is to express the coefficients a i in (2.3) by way of A i in W ( x, y ). This can bedone by comparing the coefficients of the both sides in (1.1). Our method is similar to that ofNishimura [11]. The result is the following (here, α d + i = A d + i / ( q − (cid:0) nd + i (cid:1) ): a = α d ,a = ( d − q ) α d + α d +1 ,a = 12 d ( d − q + 1) α d + ( d − q + 1) α d +1 + α d +2 ,a = 16 d ( d + 1)( d − q + 2) α d + 12 ( d + 1)( d − q + 2) α d +1 + ( d − q + 2) α d +2 + α d +3 . The coefficient a is expressed by four parameters A d , · · · , A d +3 . By invoking the binomialmoment, the number of parameters is reduced to three. In fact, we have the following: Lemma 2.2
Let W ( x, y ) be a self-dual weight enumerator of the form (1.3) and we assume g = 3 . Then we have d +3 X i = d +1 A i (cid:18) d + 4 − id + 1 (cid:19) = q d +1 X i =0 A i (cid:18) d + 4 − id + 3 (cid:19) . (2.4) Proof.
The equalities satisfied by the binomial moment of W ( x, y ) is given by n − j X i =0 (cid:18) n − ij (cid:19) A i = q n − j j X i =0 (cid:18) n − in − j (cid:19) A i ( j = 0 , , · · · , n ) (2.5)4see [10, p.131, Problem (6)]). We get (2.4) by putting n = 2 d + 4 and j = d + 1.Using A = 1, A = · · · = A d − = 0, we can see that (2.4) gives a linear relation among A d , · · · , A d +3 , so we can express A d +3 by A d , A d +1 and A d +2 . Thus we get a = −
12 ( d + 1)( dq + d − q + 2) α d − ( qd + d + 2) α d +1 − ( q + 1) α d +2 + 1 . Rewriting (2.3) using above a i , we obtain Theorem 1.6. We examine the polynomials (1.8), which has essentially only one parameter q and is easy to seethe phenomenon. Using Nishimura’s results ( g = 1 ,
2) and our theorem ( g = 3), we can see thatthe range of q for which the Riemann hypothesis is true are the following: g = 1 : 4 − √ ≈ . ≤ q ≤ √ ≈ . q = 1) ,g = 2 : − √ ≈ . ≤ q ≤ α ( ≈ . q = 1) , where α = 16 (cid:18) q √
6) + q − √ (cid:19) , and g = 3 : β ( ≈ . ≤ q ≤ β ( ≈ . q = 1) , (3.1)where β is the unique real root of the polynomial100 t + 495 t + 2056 t − t + 1408 t − β is the positive root of the polynomial13 t + 4 t − t − t − . The cases g = 1 and 2 are not very complicated, but the last case needs some explanation. Therelevant coefficients of W ,q ( x, y ) are A d = A = 4( q − , A = 0 , A = 6( q − . Using these values, we get the explicit form of the polynomial (1.7) as follows: g ( X ) := 5 X + 5( q − X − q − X − q + 20 q − . (3.2)Let D g be the discriminant of g ( X ), X and X be the roots of g ′ ( X ) (we assume X , X arereal and X ≤ X ). Then, by Theorem 1.6, W ,q ( x, y ) satisfies the Riemann hypothesis if andonly if D g ≥ , − √ q ≤ X , X ≤ √ q,g ( − √ q ) ≤ , g (2 √ q ) ≥ .
5e have D g
35 = 100 q + 495 q + 2056 q − q + 1408 q − , so D g ≥ q ≥ β (3.3)with the above mentioned β . The roots X i are given by X = − q − − p q + 230 q − ,X = − q −
2) + p q + 230 q − . The range of q satisfying − √ q ≤ X is (note that we also have 25 q + 230 q − ≥ √ − ≤ q ≤ β , (3.4)where β is the square of the unique real root of the polynomial10 t − t − t − − √ q = X ). The explicit value is β = 1300 (cid:18)
761 + q √ q − √ (cid:19) ( β ≈ . β can be obtained by constructing the cubic polynomial havingthe squares of roots of (3.5) as its roots: 100 t − t + 172 t − X ≤ √ q gives ( √ − / ≤ q . Finally, putting √ q = t , we have g ( − √ q ) = 13 t + 4 t − t − t − ,g (2 √ q ) = 13 t − t − t + 24 t − . The inequalities g ( − √ q ) ≤ g (2 √ q ) ≥ ≤ q ≤ β and q ≥ β ≈ . , (3.6)respectively. Gathering the inequalities (3.3), (3.4) and (3.6), we obtain the estimate (3.1).We can see from the above estimation that the range of q for which the Riemann hypothesisis true becomes smaller as n becomes larger. We show some results of numerical experimentfor W n,q ( x, y ). In the following table, “RH true” means the range of n where the Riemannhypothesis for W n,q ( x, y ) seems to be true: q RH true2 2 ≤ q ≤ ≤ n ≤ ≤ n ≤ ≤ n ≤ ≤ n ≤ ≤ n ≤ Conjecture 3.1
For any n ≥ , there exists q ( q ≈ ) and W n,q ( x, y ) satisfies the Riemannhypothesis. References [1] K. Chinen: Zeta functions for formal weight enumerators and the extremal property, Proc.Japan Acad. 81 Ser. A. (2005), 168-173.[2] K. Chinen: An abundance of invariant polynomials satisfying the Riemann hypothesis,Discrete Math. 308 (2008), 6426-6440.[3] K. Chinen: Construction of divisible formal weight enumerators and extremal polynomialsnot satisfying the Riemann hypothesis, arXiv:1709.03380.[4] K. Chinen: Extremal invariant polynomials not satisfying the Riemann hypothesis,arXiv:1709.03389, to appear in Appl. Algebra Engrg. Comm. Comput.[5] K. Chinen: On some families of divisible formal weight enumerators and their zeta functions,arXiv:1709.03396.[6] I. Duursma: Weight distribution of geometric Goppa codes, Trans. Amer. Math. Soc. 351,No.9 (1999), 3609-3639.[7] I. Duursma: From weight enumerators to zeta functions, Discrete Appl. Math. 111 (2001),55-73.[8] I. Duursma: A Riemann hypothesis analogue for self-dual codes, DIMACS series in DiscreteMath. and Theoretical Computer Science 56 (2001), 115-124.[9] I. Duursma: Extremal weight enumerators and ultraspherical polynomials, Discrete Math.268, No.1-3 (2003), 103-127.[10] F. J. MacWilliams, N. J. A. Sloane: The Theory of Error-Correcting Codes, North-Holland,Amsterdam, 1977.[11] S. Nishimura: On a Riemann hypothesis analogue for selfdual weight enumerators of genusless than 3, Discrete Appl. Math.156