Primitive Quantum BCH Codes over Finite Fields Salah A. Aly, Andreas Klappenecker, Pradeep Kiran Sarvepalli Department of Computer Science Texas A&M University College Station, TX 77843-3112,USA Email: {salah,klappi,pradeep}@cs.tamu.edu Abstract—An attractive feature of BCH codes is that one of the associated primitive, narrow-sense BCH code C. This 6 can infer valuable information from their design parameters entails the following tasks: 0 (length, size of the finite field, and designed distance), such as 0 bounds on the minimum distance and dimension of the code. a) Determine the design parameters for which C⊥ ⊆C; 2 In this paper, it is shown that one can also deduce from the b) determine the dimension of C; designparameterswhetherornotaprimitive,narrow-senseBCH c) bound the minimum weight in C\C⊥. n containsitsEuclideanor Hermitian dualcode. Thisinformation a is invaluable in theconstruction of quantumBCH codes. A new Incase q isaperfectsquare,wewouldalsoliketoanswerthe J proof is provided for the dimension of BCH codes with small Hermitian versions of questions a) and c): 8 designed distance, and simple bounds on the minimum distance a’) Determine the design parameters for which C⊥h ⊆C; 1 of such codes and their duals are derived as a consequence. These results allow us to derive the parameters of two families c’) bound the minimum weight in C\C⊥h. 2 of primitive quantum BCH codes as a function of their design v parameters. To putourwork into perspective,we sketch our resultsand 6 give a brief overview of related work. 2 I. INTRODUCTION Let C be a primitive, narrow-sense BCH code C of length 11 Let α denote a primitive element in the finite field Fqm. n=qm−1, m≥2, over Fq with designed distance δ. 0 We set n = qm−1 and denote by δ an integer in the range Toanswerquestiona),weproveinTheorem2thatC⊥ ⊆C 5 2 ≤ δ ≤ n. Recall that a cyclic code of length n over Fq holds if and only if δ ≤ q⌈m/2⌉ −1−(q −2)[m odd]. The 0 is called a primitive, narrow-sense BCH code with designed significanceofthisresultisthatallowsonetoidentifyallBCH h/ distance δ if its generator polynomial is of the form codes that can be used in the quantum code construction 1). p Fortunately,this question can be answered now without com- g(x)= (x−αz) with Z =C ∪···∪C , - 1 δ−1 putations.Steaneprovedin[12] thespecialcase q =2,which nt zY∈Z is easier to show, since in this case there is no difference a where Cx = {xqk modn|0 ≤ k < m} denotes the q-ary between even and odd m. u q cyclotomic coset of x modulo n. We refer to such a code as InTheorem4,weanswerquestiona’)andshowthatC⊥h ⊆ : a BCH(n,q;δ) code, and call Z the defining set of the code. C if and only if δ ≤ q(m+[meven])/2−1−(q−2)[m even], v Thebasicpropertiesoftheseclassicalcodesarediscussed,for wherewe assumethatq is a perfectsquare.Thisresultallows i X example, in the books [9], [10], [11]. us to determine all primitive, narrow-sense BCH codes that r Given a classical BCH code, we can use one of the follow- can be used in construction2). We are not aware of any prior a ing well-known constructions to derive a quantum stabilizer work concerning the Hermitian case. code: In the binary case, an answer to question b) was given 1) If there exists a classical linear [n,k,d]q code C such by MacWilliams and Sloane [11, Chapter 9, Corollary 8]. that C⊥ ⊆ C, then there exists an [[n,2k−n,≥ d]]q Apparently, Yue and Hu answered question b) in the case stabilizercodethatispuretod.Iftheminimumdistance of small designed distances [15]. We give a new proof of ofC⊥ exceedsd,thenthequantumcodeispureandhas this result in Theorem 7 and show that the dimension k = minimum distance d. n − m⌈(δ−1)(1−1/q)⌉ for δ in the range 2 ≤ δ < 2) If there exists a classical linear [n,k,d]q2 code D such q⌈m/2⌉+1. As a consequence of our answer to b), we obtain that D⊥h ⊆ D, then there exists an [[n,2k−n,≥ d]]q the dimensions of the quantum codes in constructions 1) and stabilizercodethatispuretod.Iftheminimumdistance 2). of D⊥h exceeds d, then the quantum code is pure and FindingthetrueminimumdistanceofBCHcodesisanopen has minimum distance d. problem for which a complete answer seems out of reach, The orthogonalityrelationsare definedin the Notationsat the see [3]. As a simple consequence of our answer to b), we end ofthis section. Examplesof certainbinaryquantumBCH obtainbetter boundson the minimumdistance for some BCH codes have been given in [2], [5], [7], [13]. codes, and we derive simple bounds on the (Hermitian) dual Our goal is to derive the parameters of the quantum stabi- distance of BCH codes with small designed distance, which lizer code as a functionof their design parametersn, q, and δ partly answers c) and c’). InSectionV,alltheseresultsareusedtoderivetwofamilies If y¯=n−x, then y¯=y¯ +y¯ q+···+y¯ qm−1 = 0 1 m−1 ofquantumBCH codes.Impatientreadersshouldnowbrowse m−1(q−1−x )qi. Set y = minC . We note that i=0 i −x this section to get the bigger picture. Theorem 12 yields the Py is a conjugate of y¯. Thus, the digits of y are obtained result that one obtains using construction 1). The result of by cyclically shifting the digits of y¯. construction 2) is given in Theorem 13. 3a) First we consider the case when m is even. Then the q-ary expansion of x has at least m/2 zero digits. Notations. We denote the ring of integers by Z and a finite Therefore, at least m/2 of the y¯ are equal to q −1. field with q elements by F . We follow Knuth and attribute i q Thus, y ≥ m/2−1(q−1)qi =qm/2−1=δ . to [P(k)] the value 1 if the property P(k) of the integer k i=0 max 3b) If m is oddP,then as1≤x1 and y¯=y¯ +y¯ q+···+(y¯ )q(m−1)/2+ k−1mod2, but the left hand side seems more readable. If 0 1 (m−1)/2 x and y are vectors in Fn, then we write x ⊥ y if and only (q − 1)q(m+1)/2 + ··· + (q − 1)qm−1. For 0 ≤ j ≤ if x·y =0. Similarly, if xq and y are vectors in Fn , then we (m−1)/2, we observe that xqj < n, and since y¯qj ≡ q2 −xqj modn,y¯qj = n−xqj ≥ qm−1−(q(m+1)/2− write x⊥ y if and only if xq ·y =0. h q)q(m−1)/2 = q(m+1)/2−1 ≥ δ . For (m+1)/2 ≤ max II. EUCLIDEAN DUAL CODES j ≤m−1, we find that Recall that the Euclidean dual code C⊥ of a code C ⊆Fnq y¯qj modn=y¯ +···+y¯ qj−(m+1)/2 is given by C⊥ ={y ∈Fn|x·y =0 for all x∈C}. Steane m−j (m−1)/2 q +(q−1)qj−(m−1)/2+···+(q−1)qj−1 showed in [12] that a primitive binary BCH code of length 2m −1 contains its dual if and only if its designed distance +y¯ qj +···+y¯ qm−1, 0 m−j−1 δ satisfies δ ≤2⌈m/2⌉−1. In this section we derive a similar ≥(q(m−1)/2−1)qj−(m−1)/2+y¯ +··· 0 condition for nonbinary BCH codes. +y¯ , Lemma 1: Suppose that gcd(n,q) = 1. A cyclic code of (m−1)/2 length n over Fq with defining set Z contains its Euclidean ≥q(m+1)/2−q+1=δmax, dualcodeifandonlyifZ∩Z−1 =∅, whereZ−1 denotesthe wherey¯ +···+y¯ ≥1 becausex δmax. Seeking a denotestheconjugateofthevectory =(y1,...,yn).Thegoal contradiction,weassumethatthedefiningsetZ contains of this section is to establish when a primitive, narrow-sense the set {1,...,s}, where s = δmax. By Lemma 1, BCH code contains its Hermitian dual code. it suffices to show that Z ∩ Z−1 is not empty. If Lemma 3: Assume that gcd(n,q) = 1. A cyclic code of m is even, then s = qm/2 − 1, and Z−1 contains length n over Fq2 with defining set Z contains its Hermitian the element −sqm/2 ≡ qm/2 − 1 ≡ smodn, which dual code if and only if Z ∩ Z−q = ∅, where Z−q = means that Z ∩Z−1 6= ∅; contradiction. If m is odd, {−qz modn|z ∈Z}. then s = q(m+1)/2 − q + 1, and the element given Proof: Let N ={0,1,...,n−1}. If g(z)= (z− by −sq(m−1)/2 ≡ q(m+1)/2 −q(m−1)/2 −1modn is αx) is the generator polynomial of a cyclic codQexC∈,Zthen contained in Z−1. Since this element is less than s h†(z) = (z −α−qx) is the generator polynomial of for m ≥ 3, it is contained in Z, so Z ∩ Z−1 6= C⊥h. ThuQs,xC∈N⊥\hZ⊆C if and only if g(z) divides h†(z). The ∅; contradiction. Combining these two cases, we can latterconditionisequivalenttoZ ⊆{−qx|x∈N\Z},which conclude that δ ≤ q⌈m/2⌉ −1−(q−2)[m is odd] for can also be expressed as Z∩Z−q =∅. m≥2. Theorem 4: A primitive,narrow-senseBCH codeoflength 2) ZFo−r1th=e c∅o,nvwehrsiceh, wimepplrioevseCth⊥at⊆if δC≤byδmLaxe,mtmhean1Z. ∩It q2m−1 over Fq2, where m6=2, contains its Hermitian dual code if and only if its designed distance δ satisfies suffices to show that minC ≥δ for any coset C −x max x in Z. Since 1 ≤x < δ ≤q⌈m/2⌉−1, we can write δ ≤δ =qm+[meven]−1−(q2−2)[m even]. max max x as a q-ary integer of the form x= x +x q+···+ Proof: Let n =q2m−1. Recall that the defining set Z 0 1 x qm−1with0≤x δ . Ifmiseven,thenj ≤m/2;thus,x≥qm/2+1.Ifmisodd, max Seekingacontradiction,we assumethatthedefiningset thenj ≤m/3anditfollowsthatx≥(qm−1)/(qm/3−1),and Z contains {1,...,s}, where s = δ . By Lemma 3, it is easy to see that the latter term is larger than q⌈m/2⌉+1. max it suffices to show that Z ∩Z−q is not empty. If m is In both cases this contradicts our assumption that 1 ≤ x ≤ odd, then s = qm −1. Notice that n−qsq2(m−1)/2 = q⌈m/2⌉; hence |Cx|=m. qm −1 = s, which means that s ∈ Z ∩Z−q, and this Lemma 6: Letqbeapowerofaprime,mapositiveinteger, contradictsourassumptionthatthissetisempty.Ifmis and n = qm −1. Let x and y be integers in the range 1 ≤ even,thens=qm+1−q2+1.Wenotethatn−qsqm−2 = x,y 2. It q-ary cosets of x and y modulo n are disjoint, i.e., Cx 6=Cy. followsthatqm+1−qm−1−1∈Z∩Z−q, contradicting Proof: Seeking a contradiction, we assume that Cx = our assumption that this set is empty. Combining the Cy. This assumption implies that y ≡ xqℓ modn for some two cases, we can concludethat s must be smaller than integer ℓ in the range 1≤ℓ 1, then |C | < m implies that there oflengthn=42−1=15overF .Ifwechoosethedesigned x 4 distanceδ =6>41+1,thentheresultingcodehasdimension Lemma 11: Suppose that C is a primitive, narrow-sense k =8, because the defining set Z is given by BCH code of length n = q2m −1 over Fq2 with designed distance2≤δ ≤δ =qm+[meven]−1−(q2−2)[m even]), max Z =C1∪C2∪···∪C5 ={1,4}∪{2,8}∪{3,12}∪{5}. then the dual distance d⊥ ≥δ +1. max Thedimensionformula(1)yields42−1−2⌈(6−1)(1−1/4)⌉= Proof: The proof is analogousto the one of Lemma 10; just keep in mind that the defining set Z is invariant under 7,sotheformuladoesnotextendbeyondtherangeofdesigned δ multiplication by q2 modulo n. distances given in Theorem 7. V. FAMILIESOF QUANTUM CODES B. Distance Bounds We use the results of the previous sections to prove the The true minimum distance d of a primitive BCH code min existence of quantum stabilizer codes. over F with designed distance δ is bounded by δ ≤d ≤ q min Theorem 12: If q is a power of a prime, and m and δ are qδ−1,see[11,p.261].IfweapplytheFarrbound(essentially integers such that m ≥ 2 and 2 ≤ δ ≤ δ = q⌈m/2⌉−1− the sphere packing bound) using the dimension given in max (q−2)[m odd], then there exists a quantumstabilizer code Q Theorem 7, then we obtain: with parameters Corollary 9: If C is primitive, narrow-sense BCH code of length qm−1 over F with designed distance δ in the range [[qm−1,qm−1−2m⌈(δ−1)(1−1/q)⌉,d ≥δ]] q Q q 2≤δ ≤q⌈m/2⌉+1 such that thatispureuptoδ +1.IfBCH(n,q;δ)hastrueminimum max ⌊(δ+1)/2⌋ qm−1 distanced,andd≤δmax,thenQisapurequantumcodewith (q−1)i >qm⌈(δ−1)(1−1/q)⌉, (2) minimum distance d =d. (cid:18) i (cid:19) Q Xi=0 Proof: Theorem 7 and 2 imply that there exists a then C hasminimumdistance d=δ or δ+1; if, furthermore, classicalBCH codewithparameters[qm−1,qm−1−m⌈(δ− δ ≡0modq, then d=δ+1. 1)(1−1/q)⌉,≥δ]q whichcontainsitsdualcode.An[n,k,d]q Proof: Seeking a contradiction, we assume that the code that contains its dual code implies the existence of the minimum distance d of the code satisfies d ≥ δ + 2. We quantumcode with parameters[[n,2k−n,≥d]]q by the CSS know from Theorem 7 that the dimension of the code is construction, see [7], [5]. By Lemma 10, the dual distance k = qm − 1 − m⌈(δ − 1)(1 − 1/q)⌉. If we substitute this exceeds δmax; the statement about the purity and minimum value of k into the sphere-packing bound distance is an immediate consequence. Theorem 13: If q is a power of a prime, m is a positive ⌊(d−1)/2⌋ qm−1 integer,and δ is an integer in the range 2≤δ ≤qm−1, then qk (q−1)i ≤qn, (cid:18) i (cid:19) there exists a quantum code Q with parameters Xi=0 [[q2m−1,q2m−1−2m⌈(δ−1)(1−1/q2)⌉,d ≥δ]] then we obtain Q q ⌊(δ+1)/2⌋ qm−1 ⌊(d−1)/2⌋ qm−1 that is pure up to δmax + 1, where δmax = qm+[meven] − (q−1)i ≤ (q−1)i 1−(q2 −2)[m even]). If BCH(n,q2;δ) has true minimum (cid:18) i (cid:19) (cid:18) i (cid:19) Xi=0 Xi=0 distance d, with d≤δmax, then Q is a pure quantum code of ≤qm⌈(δ−1)(1−1/q)⌉, minimum distance dQ =d. Proof: It follows from Theorems 7 and 4 that there but this contradicts condition (2); hence, δ ≤d≤δ+1. exists a primitive, narrow-sense [q2m−1,q2m−1−m⌈(δ− If δ ≡0modq, then the cyclotomic coset Cδ is contained 1)(1−1/q2)⌉,≥ δ]q2 BCH code that contains its Hermitian in the defining set Z of the code because C = C . Thus, δ δ/q dual code. Recall that if a classical [n,k,d]q2 code C exists the BCH bound implies that the minimum distance must be that contains its Hermitian dual code, then there exists an at least δ+1. [[n,2k−n,≥d]] quantumcode that is pure up to d, see [1]; q Lemma 10: Suppose that C is a primitive, narrow-sense this proves our claim. By Lemma 11, the Hermitian dual BCH code of length n = qm − 1 over F with designed q distance exceeds δ , which implies the last statement of max distance 2 ≤ δ ≤ δ = q⌈m/2⌉ − 1 − (q − 2)[m odd]), max the claim. then the dual distance d⊥ ≥δ +1. max Proof: LetN ={0,1,...,n−1}andZ bethedefining VI. CONCLUSIONS δ set of C. We know that Zδmax ⊇ Zδ ⊃ {1,...,δ − 1}. Wehaveinvestigatedprimitive,narrow-senseBCHcodesin Therefore N \ Zδmax ⊆ N \ Zδ. Further, we know that thisnote.Wewereabletocharacterizewhenprimitive,narrow- Z∩Z−1 =∅if2≤δ ≤δmax fromLemma1andTheorem2. sense BCH codes contain their Euclidean and Hermitian dual Therefore, Zδ−m1ax ⊆N \Zδmax ⊆N \Zδ. codes, and this allowed us to derive two series of quantum Let Tδ be the defining set of the dual code. Then Tδ = stabilizer codes. These results make it possible to construct (N \Zδ)−1 ⊇ Zδmax. Moreover {0} ∈ N \Zδ and therefore more families of quantum BCH codes as shown by Cohen, Tδ. Thusthere are at least δmax consecutiverootsin Tδ. Thus Encheva, and Litsyn in [4], since the BCH codes are nested the dual distance d⊥ ≥δmax+1. and are amenable to the Steane enlargement technique [12]. From a practical point of view, it is interesting that efficient [4] G. Cohen, S. Encheva, and S. Litsyn, “On binary constructions of encodinganddecodingalgorithmsareknownforcyclicbinary quantumcodes,”IEEETrans.Inform.Theory,vol.45,no.7,pp.2495– 2498,1999. quantum stabilizer codes, see [8]. [5] M.GrasslandT.Beth,“QuantumBCHcodes,”inProc.X.Int’l.Symp. Theoretical Electrical Engineering, Magdeburg,1999,pp.207–212. ACKNOWLEDGMENT [6] M. Grassl, T. Beth, and T. Pellizzari, “Codes for the quantum erasure channel,” Phys.Rev.Lett.A,vol.56,no.1,pp.33–38,1997. We thank Pascale Charpin for sending us a copy of [3], [7] M. Grassl, T. Beth, and M. Ro¨tteler, “On optimal quantum codes,” Internat. J.Quantum Information, vol.2,no.1,pp.757–775,2004. Simon Litsyn for sending us reference [4], and Neil Sloane [8] M. Grassl and T. Beth, “Cyclic quantum error-correcting codes and for very helpful discussions. quantum shift registers,” Proc. Royal Soc. London Series A, vol. 456, This research was supported by NSF CAREER award no.2003,pp.2689–2706, 2000. [9] W.C.HuffmanandV.Pless,FundamentalsofError-CorrectingCodes. CCF 0347310, NSF grant CCR 0218582 and a Texas A&M Cambridge: University Press,2003. TITF initiative. 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Primitive Quantum BCH Codes over Finite Fields

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