Mavron, V. C.

McDonough, Thomas

English

This paper gives an explicit value for the dimension of the code of a strongly resolvable design over the field of prime order $p$ in the case when $p$ is not a divisor of $k-\rho$, where $k$ is the block size of the design and $\rho$ is the number of points in the intersection of two distinct blocks in the same resolution class

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Aberystwyth UniversityThe Dimension of the Code of a Strongly Resolvable DesignMavron, V. C.; McDonough, ThomasPublication date:2010Citation for published version (APA):Mavron, V. C., & McDonough, T. (2010). The Dimension of the Code of a Strongly Resolvable Design. 203-206.General rightsCopyright and moral rights for the publications made accessible in the Aberystwyth Research Portal (the Institutional Repository) areretained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by thelegal requirements associated with these rights.            • Users may download and print one copy of any publication from the Aberystwyth Research Portal for the purpose of private study orresearch.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the Aberystwyth Research PortalTake down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.tel: +44 1970 62 2400email: is@aber.ac.ukDownload date: 18. Apr. 2020The Dimension of the Code of a StronglyResolvable DesignT. P. McDonough V. C. MavronJanuary 9, 2010AbstractThis paper gives an explicit value for the dimension of the code of astrongly resolvable design over the field of prime order p in the case whenp is not a divisor of k − ρ, where k is the block size of the design and ρis the number of points in the intersection of two distinct blocks in thesame resolution class.Keywords: Codes, finite geometry, strongly resolvable and affine designs.2010 Mathematics Subject Classification: Primary 94B05, 05B25, 94B27; Secondary05B15, 94B651 IntroductionIn this paper we consider some situations in which the dimension of the code ofa strongly resolvable design over the field of prime order p may be determinedexplicitly. We show that there is such a form when p is not a divisor of k − ρ,where k is the block size of the design and ρ is the size of the intersection of twodistinct blocks in the same resolution class.The definitions and notation for designs that we shall need are outlinedbriefly here; more details can be found in the books by Assmus and Key [1] andby Beth, Jungnickel and Lenz [2].A t-(v, k, λ) design D, where t, v and k are positive integers, and λ is anon-negative integer, consists of a finite set P of order v, called the points of D,together with a collection B of k-subsets of P, called blocks, such that any twodistinct points are contained in exactly λ blocks.It is well-known that a t-design is also an s-design for any s with 0 < s ≤ t.The number of blocks through any point is therefore a constant and is usuallydenoted by r and called the replication number. The number of blocks |B| isdenoted by b.The design D is symmetric if b = v. A 2-design with v > k > 1 is symmetricif, and only if, any two distinct blocks meet in a constant number of points (theconstant is necessarily λ); or equivalently the dual design is also a 2-design.1The design D is resolvable if it has a resolution; that is, B can be partionedinto subsets, called parallel classes, such that each parallel class partitions P.Blocks in the same parallel class are said to be parallel. It is easy to see that thenumber of parallel classes is r and that each parallel class has the same numberm = v/k of blocks.If the resolution is such that any two non-parallel blocks meet in a constantnumber of points, say µ, then Π is said to be affine. In this case it is easy toshow that µ = k/m = k2/v.Affine 1-designs are also called nets. Some authors reserve the term net foraffine designs with µ = 1 only. Nets are also the dual designs of transversaldesigns and their existence is equivalent to that of certain types of orthogonalarrays. For more details, see Hedeyat, Sloane and Stufken [3].The concept of strong resolvability is related to that of resolvability but isnot quite a generalization of it.Let D = (P,B) be a 1-(v, k, r) design, with v > k. Then D is said to bestrongly resolvable if there exist constants ρ, σ, and µ, and a partition of B, intom-subsets called resolution classes, such that:• through any point there are exactly σ blocks from each resolution class;• any two distinct blocks meet in either ρ or µ points together, accordingas they are or are not in the same resolution class.We call ρ the inner constant and µ the outer constant.We have the following arithmetical relations between these parameters: (i) ρ =k(km − v)/v(m − 1), (ii) µ = k2/v, (iii) σ = rm/b. The number of resolutionclasses is r/σ. Some further relations are: (iv) k+ ρ(m− 1) = µm, (v) µb = rk,(vii) (k − ρ)(m − 1) = σ(v − k) = k(m − σ). Note that 0 ≤ ρ < k, 1 ≤ µ < kand 1 ≤ σ < m.Affine designs are the strongly resolvable designs with ρ = 0, σ = 1 and, ifk > 1, µ = k/m.Strongly resolvable designs with only one resolution class are precisely thedual designs of 2-designs. Therefore, symmetric 2-designs are precisely thestrongly resolvable 2-designs with only one resolution class.Let p be a prime and let Fp be the field of size p. Let V be the v-dimensionalFp-space. The code C(D) of the design D over the field Fp is the subspacegenerated by the characteristic vectors of the blocks of the design, regarded assubsets of its point set, in Fp-space. The p-dimension of C(D) is its dimensionas a subspace of V .2 The Dimension TheoremLet D = (P,B) be a strongly resolvable 1-(v, k, r) design, where k < v, withresolution classes of size m > 1, inner constant ρ and outer constant µ, and letb = |B|.Let p be a prime and let Fp be the field of size p. Let V be the Fp-spacewith basis {vP : P ∈ P} of size |P|, and let C = C(D) be the Fp-code generated2by the blocks of D, considered as characteristic vectors ∈ V ; that is, B ∈ B isconsidered as∑P∈P, P I B vP .Theorem 2.1 (The Dimension Theorem) Suppose that p 6 | k − ρ.If p 6 | σ also then C has dimension 1 + r(m− 1)/σ.If p | σ then C has dimension r(m− 1)/σ.Proof. Since each point is on σ blocks of each class, the sum of the m blocks ofany resolution class is σj, where j is the all one vector. If p 6 | σ, then j ∈ C. Ifp | σ, then the m blocks in a resolution class are linearly dependent. Therefore,C is generated by any set S = S ′ ∪ S ′′, where S ′ is any set of b − b/m vectorsconsisting of any m− 1 blocks from each of the b/m block classes, and S ′′ is {j}or ∅ according as p 6 | σ or p | σ.Letλ′j+∑v∈S′λvv = 0 (1)be an Fp-linear relation involving only the vectors in S. If p | σ, set λ′ = 0.Given any two blocks u and v, their inner product is k, or ρ, or µ, accordingas the two blocks are equal, or distinct and in the same resolution class, or indifferent resolution classes.Let T be a resolution class of D and let T ′ = T ∩ S ′. Let u ∈ T ′. Takingthe inner product with u of both sides of the (1), we getkλ′ + kλu + ρ∑v∈T ′−{u}λv + µ∑v∈S′−T ′λv = 0. (2)Thuskλ′ + ρ∑v∈T ′λv + µ∑v∈S′−T ′λv = −(k − ρ)λu. (3)Now let u′ ∈ T −T ′. Taking the inner product with u′ of both sides of the (1),we getkλ′ + ρ∑v∈T ′λv + µ∑v∈S′−T ′λv = 0. (4)From (3) and (4),(k − ρ)λu = 0. (5)Since p 6 | k − ρ, λu = 0 for all u ∈ S ′. If p | σ, we have established that therelation in (1) is trivial. If p 6 | σ, the relation in (1) becomes λ′j = 0. So λ′ = 0and the relation is trivial. This completes the proof.The proof of the following corollary is straightforward.Corollary 2.2 Suppose that D is an affine 1-(µm2, µm, r) design (i.e a net)and that p 6 | µm. Then C has dimension 1 + r(m− 1).The case when p divides k−ρ is more difficult, though it is the more interestingcase for coding theorists. The Smith normal form of the incidence matrix of thedesign may give some insight as to why the cases differ. For example, consider3the case of the unique net with µ = 2 and m = 4. (We are indebted to VladimirTonchev for the information that there is only one net with these parametersand for computing the Smith normal form of its 32× 32 incidence matrix.) Theinvariant factors of the incidence matrix consist of 13 ones, 4 twos, 6 fours and2 eights.This explains why the p-dimension of the code of this net is 13 if p = 2 andis 25 otherwise.The latter number agrees with that given by the corollary.References[1] E. F. Assmus, Jr and J. D. Key, Designs and their Codes, Cambridge: Cam-bridge University Press, 1992, Cambridge Tracts in Mathematics, Vol. 103(Second printing with corrections, 1993).[2] T. Beth, D. Jungnickel and H. Lenz, Designs and their Codes, Cam-bridge: Cambridge University Press, 1992, Cambridge Tracts in Mathe-matics, Vol. 103 (Second printing with corrections, 1993).[3] Orthogonal Arrays, New York: Springer-Verlag, 1999.Institute of Mathematics and Physics, Aberystwyth University,Aberystwyth, Ceredigion SY23 3BZ, U.K.E-mail: tpd@aber.ac.ukInstitute of Mathematics and Physics, Aberystwyth University,Aberystwyth, Ceredigion SY23 3BZ, U.K.E-mail: vcm@aber.ac.uk4

The Dimension of the Code of a Strongly Resolvable Design

Aberystwyth UniversityThe Dimension of the Code of a Strongly Resolvable DesignMavron, V. C.; McDonough, ThomasPublication date:2010Citation for published version (APA):Mavron, V. C., & McDonough, T. (2010). The Dimension of the Code of a Strongly Resolvable Design. 203-206.http://hdl.handle.net/2160/5967General rightsCopyright and moral rights for the publications made accessible in the Aberystwyth Research Portal (the Institutional Repository) areretained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by thelegal requirements associated with these rights.            • Users may download and print one copy of any publication from the Aberystwyth Research Portal for the purpose of private study orresearch.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the Aberystwyth Research PortalTake down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.tel: +44 1970 62 2400email: is@aber.ac.ukDownload date: 09. Jul. 2020The Dimension of the Code of a StronglyResolvable DesignT. P. McDonough V. C. MavronJanuary 9, 2010AbstractThis paper gives an explicit value for the dimension of the code of astrongly resolvable design over the field of prime order p in the case whenp is not a divisor of k − ρ, where k is the block size of the design and ρis the number of points in the intersection of two distinct blocks in thesame resolution class.Keywords: Codes, finite geometry, strongly resolvable and affine designs.2010 Mathematics Subject Classification: Primary 94B05, 05B25, 94B27; Secondary05B15, 94B651 IntroductionIn this paper we consider some situations in which the dimension of the code ofa strongly resolvable design over the field of prime order p may be determinedexplicitly. We show that there is such a form when p is not a divisor of k − ρ,where k is the block size of the design and ρ is the size of the intersection of twodistinct blocks in the same resolution class.The definitions and notation for designs that we shall need are outlinedbriefly here; more details can be found in the books by Assmus and Key [1] andby Beth, Jungnickel and Lenz [2].A t-(v, k, λ) design D, where t, v and k are positive integers, and λ is anon-negative integer, consists of a finite set P of order v, called the points of D,together with a collection B of k-subsets of P, called blocks, such that any twodistinct points are contained in exactly λ blocks.It is well-known that a t-design is also an s-design for any s with 0 < s ≤ t.The number of blocks through any point is therefore a constant and is usuallydenoted by r and called the replication number. The number of blocks |B| isdenoted by b.The design D is symmetric if b = v. A 2-design with v > k > 1 is symmetricif, and only if, any two distinct blocks meet in a constant number of points (theconstant is necessarily λ); or equivalently the dual design is also a 2-design.1The design D is resolvable if it has a resolution; that is, B can be partionedinto subsets, called parallel classes, such that each parallel class partitions P.Blocks in the same parallel class are said to be parallel. It is easy to see that thenumber of parallel classes is r and that each parallel class has the same numberm = v/k of blocks.If the resolution is such that any two non-parallel blocks meet in a constantnumber of points, say µ, then Π is said to be affine. In this case it is easy toshow that µ = k/m = k2/v.Affine 1-designs are also called nets. Some authors reserve the term net foraffine designs with µ = 1 only. Nets are also the dual designs of transversaldesigns and their existence is equivalent to that of certain types of orthogonalarrays. For more details, see Hedeyat, Sloane and Stufken [3].The concept of strong resolvability is related to that of resolvability but isnot quite a generalization of it.Let D = (P,B) be a 1-(v, k, r) design, with v > k. Then D is said to bestrongly resolvable if there exist constants ρ, σ, and µ, and a partition of B, intom-subsets called resolution classes, such that:• through any point there are exactly σ blocks from each resolution class;• any two distinct blocks meet in either ρ or µ points together, accordingas they are or are not in the same resolution class.We call ρ the inner constant and µ the outer constant.We have the following arithmetical relations between these parameters: (i) ρ =k(km − v)/v(m − 1), (ii) µ = k2/v, (iii) σ = rm/b. The number of resolutionclasses is r/σ. Some further relations are: (iv) k+ ρ(m− 1) = µm, (v) µb = rk,(vii) (k − ρ)(m − 1) = σ(v − k) = k(m − σ). Note that 0 ≤ ρ < k, 1 ≤ µ < kand 1 ≤ σ < m.Affine designs are the strongly resolvable designs with ρ = 0, σ = 1 and, ifk > 1, µ = k/m.Strongly resolvable designs with only one resolution class are precisely thedual designs of 2-designs. Therefore, symmetric 2-designs are precisely thestrongly resolvable 2-designs with only one resolution class.Let p be a prime and let Fp be the field of size p. Let V be the v-dimensionalFp-space. The code C(D) of the design D over the field Fp is the subspacegenerated by the characteristic vectors of the blocks of the design, regarded assubsets of its point set, in Fp-space. The p-dimension of C(D) is its dimensionas a subspace of V .2 The Dimension TheoremLet D = (P,B) be a strongly resolvable 1-(v, k, r) design, where k < v, withresolution classes of size m > 1, inner constant ρ and outer constant µ, and letb = |B|.Let p be a prime and let Fp be the field of size p. Let V be the Fp-spacewith basis {vP : P ∈ P} of size |P|, and let C = C(D) be the Fp-code generated2by the blocks of D, considered as characteristic vectors ∈ V ; that is, B ∈ B isconsidered as∑P∈P, P I B vP .Theorem 2.1 (The Dimension Theorem) Suppose that p 6 | k − ρ.If p 6 | σ also then C has dimension 1 + r(m− 1)/σ.If p | σ then C has dimension r(m− 1)/σ.Proof. Since each point is on σ blocks of each class, the sum of the m blocks ofany resolution class is σj, where j is the all one vector. If p 6 | σ, then j ∈ C. Ifp | σ, then the m blocks in a resolution class are linearly dependent. Therefore,C is generated by any set S = S ′ ∪ S ′′, where S ′ is any set of b − b/m vectorsconsisting of any m− 1 blocks from each of the b/m block classes, and S ′′ is {j}or ∅ according as p 6 | σ or p | σ.Letλ′j+∑v∈S′λvv = 0 (1)be an Fp-linear relation involving only the vectors in S. If p | σ, set λ′ = 0.Given any two blocks u and v, their inner product is k, or ρ, or µ, accordingas the two blocks are equal, or distinct and in the same resolution class, or indifferent resolution classes.Let T be a resolution class of D and let T ′ = T ∩ S ′. Let u ∈ T ′. Takingthe inner product with u of both sides of the (1), we getkλ′ + kλu + ρ∑v∈T ′−{u}λv + µ∑v∈S′−T ′λv = 0. (2)Thuskλ′ + ρ∑v∈T ′λv + µ∑v∈S′−T ′λv = −(k − ρ)λu. (3)Now let u′ ∈ T −T ′. Taking the inner product with u′ of both sides of the (1),we getkλ′ + ρ∑v∈T ′λv + µ∑v∈S′−T ′λv = 0. (4)From (3) and (4),(k − ρ)λu = 0. (5)Since p 6 | k − ρ, λu = 0 for all u ∈ S ′. If p | σ, we have established that therelation in (1) is trivial. If p 6 | σ, the relation in (1) becomes λ′j = 0. So λ′ = 0and the relation is trivial. This completes the proof.The proof of the following corollary is straightforward.Corollary 2.2 Suppose that D is an affine 1-(µm2, µm, r) design (i.e a net)and that p 6 | µm. Then C has dimension 1 + r(m− 1).The case when p divides k−ρ is more difficult, though it is the more interestingcase for coding theorists. The Smith normal form of the incidence matrix of thedesign may give some insight as to why the cases differ. For example, consider3the case of the unique net with µ = 2 and m = 4. (We are indebted to VladimirTonchev for the information that there is only one net with these parametersand for computing the Smith normal form of its 32× 32 incidence matrix.) Theinvariant factors of the incidence matrix consist of 13 ones, 4 twos, 6 fours and2 eights.This explains why the p-dimension of the code of this net is 13 if p = 2 andis 25 otherwise.The latter number agrees with that given by the corollary.References[1] E. F. Assmus, Jr and J. D. Key, Designs and their Codes, Cambridge: Cam-bridge University Press, 1992, Cambridge Tracts in Mathematics, Vol. 103(Second printing with corrections, 1993).[2] T. Beth, D. Jungnickel and H. Lenz, Designs and their Codes, Cam-bridge: Cambridge University Press, 1992, Cambridge Tracts in Mathe-matics, Vol. 103 (Second printing with corrections, 1993).[3] Orthogonal Arrays, New York: Springer-Verlag, 1999.Institute of Mathematics and Physics, Aberystwyth University,Aberystwyth, Ceredigion SY23 3BZ, U.K.E-mail: tpd@aber.ac.ukInstitute of Mathematics and Physics, Aberystwyth University,Aberystwyth, Ceredigion SY23 3BZ, U.K.E-mail: vcm@aber.ac.uk4

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The Dimension of the Code of a Strongly Resolvable Design

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