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Orthogonality Matrices for Modules over Finite Frobenius Rings and MacWilliams' Equivalence Theorem
Finite Fields and Their Applications 8, 323–331 (2002) doi:10.1006/ffta.2001.0343
Orthogonality Matrices for Modules over Finite Frobenius Rings and MacWilliams’ EquivalenceTheorem Marcus Greferath Department of Mathematics San Diego State University, San Diego, California, 92182 E-mail: [email protected] Communicated by Vera Pless Received September 11, 2000; revised March 27, 2001; published online February 26, 2002
MacWilliams’ equivalence theorem states that Hamming isometries between linear codes extend to monomial transformations of the ambient space. One of the most elegant proofs for this result is due to K. P. Bogart et al. (1978, Inform. and Control 37, 19–22) where the invertibility of orthogonality matrices of finite vector spaces is the key step. The present paper revisits this technique in order to make it work in the context of linear codes over finite Frobenius rings. # 2002 Elsevier Science (USA) Key Words: linear codes over rings; Frobenius rings; MacWilliams’ equivalence theorem; homogeneous weights; Mo. bius inversion on posets.
0.
INTRODUCTION
The classical notion of code equivalence is based on a theorem by F. J. MacWilliams [9] stating that Hamming isometries between linear codes over finite fields can be extended to monomial transformations of the ambient vector space. This theorem has intensively been reexamined in different papers; in particular, the approach by Bogart et al. [2] contains a most elegant and amazing technique of proof. Linear codes over finite rings have been gaining importance since the end of the eighties (cf. [7,10]). A most important foundational result by Wood [13] states that the above equivalence theorem holds for linear codes over a very large class of rings, namely the finite Frobenius rings. Wood proves his result using character theoretic methods. The paper [5] contains the first purely combinatorial proof for Wood’s result, involving what is called a 323 1071-5797/02 $35.00 # 2002 Elsevier Science (USA) All rights reserved.
324
MARCUS GREFERATH
homogeneous weight on finite rings and Mo. bius inversion on partially ordered sets. The present article’s objective is to demonstrate how the method in Bogart et al. [2] can be applied to reprove the theorem in question for linear codes over finite Frobenius rings. It turns out again that homogeneous weights play a central role in the analysis of the behaviour of orthogonality matrices. We will introduce two particular kinds of these matrices and study the relation between them. Proving that these matrices are invertible over Q, we finally end up with an adapted version of the proof by Bogart et al.
1. PREPARATION Let us briefly recall the general principle of Mo. bius inversion; the reader interested in a more detailed survey may refer [1, Chap. IV; 11, 12, Chap. 3.6]. For a finite poset P , define the incidence algebra AðP Þ :¼ ff : P P ! Q j x 6 y implies f ðx; yÞ ¼ 0g; which is indeed a Q-algebra where the vector space structure is given by ðf þ gÞðx; yÞ :¼ f ðx; yÞ þ gðx; yÞ ðrf Þðx; yÞ :¼ rf ðx; yÞ for all f ; g 2 AðP Þ and r 2 Q, and where the multiplication is given by the convolution X ðf * gÞðx; yÞ :¼ f ðx; zÞgðz; yÞ x z y
for all f ; g 2 AðP Þ. Here the Kronecker function ( 1; d : P P ! Q; ðx; yÞ/ 0;
x¼y otherwise
is the left and right identity. The invertible elements are exactly the functions f 2 AðP Þ satisfying f ðx; xÞ=0 for all x 2 P . In particular the so-called zetafunction i.e., the characteristic function of the partial order of P given by ( 1; x y z : P P ! Q; ðx; yÞ/ 0; otherwise
MACWILLIAMS’ EQUIVALENCE THEOREM
325
is an invertible element of AðP Þ, and its inverse is given by the Mo. bius function m : P P ! Q implicitly defined by mðx; xÞ ¼ 1 and X mðx; tÞ ¼ 0 x t y
if x5y, and mðx; yÞ ¼ 0 if x 6 y. The Mo. bius function allows for the following equivalence: if f ; g are any rational-valued functions on P then X X gðyÞ ¼ f ðxÞ for all y 2 P , f ðyÞ ¼ gðxÞmðx; yÞ for all y 2 P : x y
x y
2. HOMOGENEOUS WEIGHTS ON FINITE RINGS Homogeneous weights were first introduced by Constantinescu [3] (cf. also [4]) when working with linear codes over integer residue rings. This class of weight functions has been revisited and generalized at different places (cf. [5, 6, 8]). Here we follow the line in [5, 6] which works without restrictions concerning the underlying class of finite rings. Definition 2.1. A weight w on the finite ring R is called left homogeneous, if wð0Þ ¼ 0 and the following is true: ðH1Þ If Rx ¼ Ry then wðxÞ ¼ wðyÞ for all x; y 2 R. ðH2Þ There exists a rational number g 0 such that X wðyÞ ¼ gjRxj for all x 2 R\f0g: y2Rx
Mo. bius inversion shows that every finite ring admits a (left) homogeneous weight (cf. [5, Theorem 1.3]): Theorem 2.2. Let R be a finite ring and let m denote the Mo. bius function on the set fRx j x 2 Rg of its principal left ideals. Left homogeneous weights on R exist and are of the form mð0; RxÞ w : R ! Q; x/g 1 ; jR xj where g is a non-negative rational constant, and R denotes the set of units of R. An essential observation in [5, Lemma 1:5, Remark 1.7(b)] is the following statement:
326
MARCUS GREFERATH
Remark 2.3. If w : R ! Q; x/gð1 mð0; RxÞ=jR xjÞ is a left homogeneous weight on the finite ring R, then R is a Frobenius ring if and only if: ðH2 * Þ For all nonzero I RR, one has
P
x2I
wðxÞ ¼ gjIj.
From Theorem 2.2, it is clear that also right associated elements on R have the same homogeneous weight. Furthermore it has been shown in Honold et al. [8] that if R is a finite Frobenius ring, then every left homogeneous weight is also right homogeneous. As it will always be clear from the context with which side of the ring we are dealing, we will simply say homogeneous instead of left homogeneous in the sequel. For all that follows denote whom as the normalized homogeneous weight on a finite ring, which means whom satisfies ðH2Þ in Definition 2.1 with g ¼ 1. The following observations from [5, Theorem 3.1, 4.1(b)] will be useful. Remark 2.4. The relationship between the Hamming weight wH and the normalized homogeneous weight whom on a finite ring R is given by wH ðzÞ ¼
1 X whom ðrzÞ jRj r2R
for all z 2 R
and equivalently, whom ðzÞ ¼
1 X jRrzj jRzj wH ðrzÞ mðRrz; RzÞ jRj r2R jR rzj jR zj
for all z 2 R:
Here m again denotes the Mo. bius function on the set of all principal left ideals of R.
3.
ORTHOGONALITY MATRICES FOR FINITE MODULES
For any nonzero finite module R M we have a natural pairing RM
MR* ! R;
ðx; yÞ/xy;
where M * :¼ HomR ðM; RÞ. This pairing is obviously linear in each component. Defining E :¼ fRx j 0=x 2 Mg and E * :¼ fyR j 0=y 2 M * g, we consider the following rational ðjEj jE * jÞ-matrices, TH :¼ ½wH ðxyÞRx2E ; yR2E *
Thom :¼ ½whom ðxyÞRx2E : yR2E *
These matrices are well defined because Rx ¼ Ry is equivalent to R x ¼ R y for every finite ring R (a short proof can be found in [13, Theorem 5.1]). At this point, it is not clear what assumptions are needed in order to guarantee
MACWILLIAMS’ EQUIVALENCE THEOREM
327
that TH and Thom are square matrices. We will see later that these matrices are square and invertible, whenever R is a Frobenius ring and R M is a free module. Note that if R is a finite field, then TH is exactly the orthogonality matrix considered by Bogart et al. [2, Sect. 2]. Example 3.1. If R ¼ Z4 the normalized homogeneous weight is given by the known Lee weight assigning 0/0; 1/1; 2/2, and 3/1. The nonzero cyclic submodules of Z24 are given by E ¼ E* ¼ fRð1; 0Þ; Rð1; 1Þ; Rð1; 2Þ; Rð1; 3Þ; Rð2; 0Þ; Rð2; 1Þ; Rð2; 2Þ; Rð0; 1Þ; Rð0; 2Þg; and we obtain (up to row matrices: 2 1 61 6 6 61 6 6 61 6 TH ¼ 6 61 6 61 6 61 6 6 40 0 2
Thom
Proposition 3.2.
and column permutations) the orthogonality 1 1
1 1 1 0
1 1
1 1
1 0 0 1
1
1 1
1
0
1 1
0 1
1 1 1 1
1 0
1 0
0 1 0 0
1 0
0 1 1 0
0 0
1 1
1 1 0 1
1 1
1 1 0 1
0 0
1 1
1 1 0 1
1 61 6 6 61 6 6 61 6 ¼6 62 6 62 6 62 6 6 40
1 2
1 1
1 0
2 2 2 1
2 0
0 1
1
1
1
2 0
2
2
0 2
1 2
2 2
2 1 0 0
0 0
1 0
1 0
0 2
1 0
0 1 0 2
2 0
1 2
1
2
1
0 1
2
1
0
2
0
2
0 2
0
2
3 0 17 7 7 07 7 7 17 7 07 7; 7 17 7 07 7 7 15 0 3 0 27 7 7 07 7 7 27 7 07 7: 7 27 7 07 7 7 25 0
TH ¼ At Thom where A is the ðjEj jEjÞ-matrix given by ARx;Ry ¼
jR xj zðRx; RyÞ: jRyj
328
MARCUS GREFERATH
The matrix A is invertible, and its inverse B is given by BRx;Ry ¼
jRxj mðRx; RyÞ: jR yj
Here z and m are the respective zeta and Mo. bius functions on the poset E. Proof. have
By Remark 2.4 and appropriate homomorphism arguments, we 1 X 1 X whom ðrxzÞ ¼ whom ðyzÞ jRj r2R jRxj y2Rx X jR yj 1 X zðRy; RxÞwhom ðyzÞ; ¼ whom ðyzÞjR yj ¼ jRxj Ry Rx jRxj Ry2E
wH ðxzÞ ¼
which shows the first part of our claim. For the second part we verify ðBAÞRx;Rz ¼
X
BRx;Ry ARy;Rz
Ry2E
X jRxj jR yj zðRy; RzÞ mðRx; RyÞ jR yj jRzj Ry2E jRxj X ¼ mðRx; RyÞ ¼ dðRx; RzÞ: jRzj Ry Rz ¼
&
Let us now show the main result which assumes the underlying ring to be a finite Frobenius ring. Theorem 3.3. If R is a finite Frobenius ring and M ¼ Rk , then Thom and hence also TH is a ðsquareÞ invertible matrix. Proof. We will introduce elementary row operations on Thom which produce an invertible ðjE * j jE * jÞ-matrix. This will in particular reveal that jEj jE * j, which by symmetry forces jEj ¼ jE * j. For arbitrary D R Rk first consider the expression ðfyR ðDÞÞyR2E * where fyR ðDÞ :¼
1 X whom ðxyÞjR xj: jDj Rx D
Being of length jE * j the vector ðfyR ðDÞÞyR2E * is a weighted sum over all rows of Thom , where the row indices range over the nonzero cyclic submodules
MACWILLIAMS’ EQUIVALENCE THEOREM
329
of D. Simple homomorphism arguments show that 1 X 1 X fyR ðDÞ ¼ whom ðx; yÞjR xj ¼ whom ðxyÞ jDj Rx D jDj x2D 1 X ¼ whom ðzÞ ¼ 1 dðDy; 0Þ; jDyj z2Dy as a consequence of Remark 2.3. Now, if D ¼ Rk , we have fyR ðDÞ ¼ 1 for all yR 2 E * . If D ¼ ðzRÞ? for some zR 2 E * , then fyR ðDÞ ¼ 1 zðyR; zRÞ: Combining this it is clear that fyR ðRk Þ fyR ððzRÞ? Þ ¼ zðyR; zRÞ; which means that we have expressed the zeta function of the partial order on E * by row combinations of Thom . Our claim now follows from the fact that the matrix representing z is invertible. & 4.
THE EQUIVALENCE THEOREM
Let a finite Frobenius ring R and an R-linear code C of length n be given. Assume that C is k-generated; i.e., there exists a ðk nÞ-generator matrix G for C. Corresponding to what we have discussed in the previous sections, we define E * :¼ fyR j 0=y 2 Rk g and consider the rational vector r :¼ ðryR ÞyR2E * where ryR :¼ jfc j c is a column of G; with cR ¼ yRgj: Clearly, r can be considered as a multiset on E * counting all nonzero columns of G up to associates. Observation 4.1 For all x 2 Rk there holds and ðThom rÞRx ¼ whom ðxGÞ: ðTH rÞRx ¼ wH ðxGÞ Proof Let w be one of wH or whom , and denote by T the corresponding orthogonality matrix on Rk . For all Rx 2 E we easily verify X ðTrÞRx ¼ wðxyÞ#fc j c is a column of G; with cR ¼ yRg yR2E *
¼
X
wðxcÞ ¼ wðxGÞ:
c2Rk c column of G
This prove our claim.
&
330
MARCUS GREFERATH
Theorem 4.2. For a finite Frobenius ring R, a left R-linear code C of length n, and an R-linear mapping C !j Rn , the following are equivalent: (a) j is a Hamming ðresp. homogeneousÞ isometry. (b) j is restriction of a monomial transformation of R Rn . Proof. The necessity of (a) for (b) is clear. Therefore assume (a) and let G be a ðk nÞ-generator matrix for C. Furthermore, let T be the respective orthogonality matrix for the underlying module M :¼ Rk . It is clear that G0 :¼ Gj (defined row-wise) is a ðk nÞ-generator matrix for C 0 :¼ Cj. For the multisets r and r0 induced by G and G0 we obtain the following: as G0 ¼ Gj we have wðxGÞ ¼ wðxG0 Þ for all x 2 C. Therefore Tr ¼ Tr0 by Observation 4.1, and hence r ¼ r0 because of Theorem 3.3. Consequently the columns of G and G0 are the same up to permutation and (right) associates, i.e., up to a monomial transformation of Rn . &
ACKNOWLEDGMENT The author thanks Sergio Lop!ez-Permouth for the opportunity to visit the Mathematics Department of Ohio University, where the results of the present paper were developed.
REFERENCES 1. M. Aigner, ‘‘Combinatorial Theory,’’ Springer-Verlag, Berlin/Heidelberg/New York, 1997. 2. K. P. Bogart, D. Goldberg, and J. Gordon, An elementary proof of the MacWilliams theorem on equivalence of codes, Inform. and Control 37 (1978), 19–22. 3. I. Constantinescu, ‘‘Lineare Codes u. ber Restklassenringen ganzer Zahlen und ihre Automorphismen bezu. glich einer verallgemeinerten Hamming-Metrik,’’ Ph.D. thesis, Technische Universit.at, Mu. nchen, 1995. 4. I. Constantinescu and W. Heise, A metric for codes over residue class rings of integers, Problemy Peredachi Informatsii 33, No. 3 (1997), 22–28. 5. M. Greferath and S. E. Schmidt, Finite-ring combinatorics and MacWilliams equivalence theorem, J. Combin. Theory, Ser. A 92 (2000), 17–28. 6. M. Greferath and S. E. Schmidt, Gray isometries for finite chain rings and a non-linear ternary ð36; 312 ; 15Þ-code, IEEE Trans. Inform. Theory 45 (1999), 2522–2524. 7. A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sol!e, The Z4 -linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory 40 (1994), 301–319. 8. T. Honold and A. A. Nechaev, Fully weighted modules and representations of codes, Problems Inform. Transmission 35 (1999), 205–223. 9. F. J. MacWilliams, ‘‘Combinatorial Properties of Elementary Abelian Groups,’’ Ph.D. thesis, Radcliffe College, Cambridge, MA, 1962.
MACWILLIAMS’ EQUIVALENCE THEOREM
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10. A. A. Nechaev, Kerdock codes in a cyclic form, Discrete Math. Appl. 1 (1991), 365–384. 11. G.-C. Rota, On the foundations of combinatorial theory. I. Theory of Mo. bius functions, Z. Wahrsch. Verw. Gebiete 2 (1964), 340–368. 12. R. P. Stanley, ‘‘Enumerative Combinatorics, Vol. 1,’’ Cambridge Univ. Press, Cambridge, UK, I997, with A foreword by Gian-Carlo Rota; corrected reprint of the 1986 original. 13. J. A. Wood, Duality for modules over finite rings and applications to coding theory, Amer. J. Math. 121 (1999), 555–575.
Orthogonality Matrices for Modules over Finite Frobenius Rings and MacWilliams’ EquivalenceTheorem Marcus Greferath Department of Mathematics San Diego State University, San Diego, California, 92182 E-mail: [email protected] Communicated by Vera Pless Received September 11, 2000; revised March 27, 2001; published online February 26, 2002
MacWilliams’ equivalence theorem states that Hamming isometries between linear codes extend to monomial transformations of the ambient space. One of the most elegant proofs for this result is due to K. P. Bogart et al. (1978, Inform. and Control 37, 19–22) where the invertibility of orthogonality matrices of finite vector spaces is the key step. The present paper revisits this technique in order to make it work in the context of linear codes over finite Frobenius rings. # 2002 Elsevier Science (USA) Key Words: linear codes over rings; Frobenius rings; MacWilliams’ equivalence theorem; homogeneous weights; Mo. bius inversion on posets.
0.
INTRODUCTION
The classical notion of code equivalence is based on a theorem by F. J. MacWilliams [9] stating that Hamming isometries between linear codes over finite fields can be extended to monomial transformations of the ambient vector space. This theorem has intensively been reexamined in different papers; in particular, the approach by Bogart et al. [2] contains a most elegant and amazing technique of proof. Linear codes over finite rings have been gaining importance since the end of the eighties (cf. [7,10]). A most important foundational result by Wood [13] states that the above equivalence theorem holds for linear codes over a very large class of rings, namely the finite Frobenius rings. Wood proves his result using character theoretic methods. The paper [5] contains the first purely combinatorial proof for Wood’s result, involving what is called a 323 1071-5797/02 $35.00 # 2002 Elsevier Science (USA) All rights reserved.
324
MARCUS GREFERATH
homogeneous weight on finite rings and Mo. bius inversion on partially ordered sets. The present article’s objective is to demonstrate how the method in Bogart et al. [2] can be applied to reprove the theorem in question for linear codes over finite Frobenius rings. It turns out again that homogeneous weights play a central role in the analysis of the behaviour of orthogonality matrices. We will introduce two particular kinds of these matrices and study the relation between them. Proving that these matrices are invertible over Q, we finally end up with an adapted version of the proof by Bogart et al.
1. PREPARATION Let us briefly recall the general principle of Mo. bius inversion; the reader interested in a more detailed survey may refer [1, Chap. IV; 11, 12, Chap. 3.6]. For a finite poset P , define the incidence algebra AðP Þ :¼ ff : P P ! Q j x 6 y implies f ðx; yÞ ¼ 0g; which is indeed a Q-algebra where the vector space structure is given by ðf þ gÞðx; yÞ :¼ f ðx; yÞ þ gðx; yÞ ðrf Þðx; yÞ :¼ rf ðx; yÞ for all f ; g 2 AðP Þ and r 2 Q, and where the multiplication is given by the convolution X ðf * gÞðx; yÞ :¼ f ðx; zÞgðz; yÞ x z y
for all f ; g 2 AðP Þ. Here the Kronecker function ( 1; d : P P ! Q; ðx; yÞ/ 0;
x¼y otherwise
is the left and right identity. The invertible elements are exactly the functions f 2 AðP Þ satisfying f ðx; xÞ=0 for all x 2 P . In particular the so-called zetafunction i.e., the characteristic function of the partial order of P given by ( 1; x y z : P P ! Q; ðx; yÞ/ 0; otherwise
MACWILLIAMS’ EQUIVALENCE THEOREM
325
is an invertible element of AðP Þ, and its inverse is given by the Mo. bius function m : P P ! Q implicitly defined by mðx; xÞ ¼ 1 and X mðx; tÞ ¼ 0 x t y
if x5y, and mðx; yÞ ¼ 0 if x 6 y. The Mo. bius function allows for the following equivalence: if f ; g are any rational-valued functions on P then X X gðyÞ ¼ f ðxÞ for all y 2 P , f ðyÞ ¼ gðxÞmðx; yÞ for all y 2 P : x y
x y
2. HOMOGENEOUS WEIGHTS ON FINITE RINGS Homogeneous weights were first introduced by Constantinescu [3] (cf. also [4]) when working with linear codes over integer residue rings. This class of weight functions has been revisited and generalized at different places (cf. [5, 6, 8]). Here we follow the line in [5, 6] which works without restrictions concerning the underlying class of finite rings. Definition 2.1. A weight w on the finite ring R is called left homogeneous, if wð0Þ ¼ 0 and the following is true: ðH1Þ If Rx ¼ Ry then wðxÞ ¼ wðyÞ for all x; y 2 R. ðH2Þ There exists a rational number g 0 such that X wðyÞ ¼ gjRxj for all x 2 R\f0g: y2Rx
Mo. bius inversion shows that every finite ring admits a (left) homogeneous weight (cf. [5, Theorem 1.3]): Theorem 2.2. Let R be a finite ring and let m denote the Mo. bius function on the set fRx j x 2 Rg of its principal left ideals. Left homogeneous weights on R exist and are of the form mð0; RxÞ w : R ! Q; x/g 1 ; jR xj where g is a non-negative rational constant, and R denotes the set of units of R. An essential observation in [5, Lemma 1:5, Remark 1.7(b)] is the following statement:
326
MARCUS GREFERATH
Remark 2.3. If w : R ! Q; x/gð1 mð0; RxÞ=jR xjÞ is a left homogeneous weight on the finite ring R, then R is a Frobenius ring if and only if: ðH2 * Þ For all nonzero I RR, one has
P
x2I
wðxÞ ¼ gjIj.
From Theorem 2.2, it is clear that also right associated elements on R have the same homogeneous weight. Furthermore it has been shown in Honold et al. [8] that if R is a finite Frobenius ring, then every left homogeneous weight is also right homogeneous. As it will always be clear from the context with which side of the ring we are dealing, we will simply say homogeneous instead of left homogeneous in the sequel. For all that follows denote whom as the normalized homogeneous weight on a finite ring, which means whom satisfies ðH2Þ in Definition 2.1 with g ¼ 1. The following observations from [5, Theorem 3.1, 4.1(b)] will be useful. Remark 2.4. The relationship between the Hamming weight wH and the normalized homogeneous weight whom on a finite ring R is given by wH ðzÞ ¼
1 X whom ðrzÞ jRj r2R
for all z 2 R
and equivalently, whom ðzÞ ¼
1 X jRrzj jRzj wH ðrzÞ mðRrz; RzÞ jRj r2R jR rzj jR zj
for all z 2 R:
Here m again denotes the Mo. bius function on the set of all principal left ideals of R.
3.
ORTHOGONALITY MATRICES FOR FINITE MODULES
For any nonzero finite module R M we have a natural pairing RM
MR* ! R;
ðx; yÞ/xy;
where M * :¼ HomR ðM; RÞ. This pairing is obviously linear in each component. Defining E :¼ fRx j 0=x 2 Mg and E * :¼ fyR j 0=y 2 M * g, we consider the following rational ðjEj jE * jÞ-matrices, TH :¼ ½wH ðxyÞRx2E ; yR2E *
Thom :¼ ½whom ðxyÞRx2E : yR2E *
These matrices are well defined because Rx ¼ Ry is equivalent to R x ¼ R y for every finite ring R (a short proof can be found in [13, Theorem 5.1]). At this point, it is not clear what assumptions are needed in order to guarantee
MACWILLIAMS’ EQUIVALENCE THEOREM
327
that TH and Thom are square matrices. We will see later that these matrices are square and invertible, whenever R is a Frobenius ring and R M is a free module. Note that if R is a finite field, then TH is exactly the orthogonality matrix considered by Bogart et al. [2, Sect. 2]. Example 3.1. If R ¼ Z4 the normalized homogeneous weight is given by the known Lee weight assigning 0/0; 1/1; 2/2, and 3/1. The nonzero cyclic submodules of Z24 are given by E ¼ E* ¼ fRð1; 0Þ; Rð1; 1Þ; Rð1; 2Þ; Rð1; 3Þ; Rð2; 0Þ; Rð2; 1Þ; Rð2; 2Þ; Rð0; 1Þ; Rð0; 2Þg; and we obtain (up to row matrices: 2 1 61 6 6 61 6 6 61 6 TH ¼ 6 61 6 61 6 61 6 6 40 0 2
Thom
Proposition 3.2.
and column permutations) the orthogonality 1 1
1 1 1 0
1 1
1 1
1 0 0 1
1
1 1
1
0
1 1
0 1
1 1 1 1
1 0
1 0
0 1 0 0
1 0
0 1 1 0
0 0
1 1
1 1 0 1
1 1
1 1 0 1
0 0
1 1
1 1 0 1
1 61 6 6 61 6 6 61 6 ¼6 62 6 62 6 62 6 6 40
1 2
1 1
1 0
2 2 2 1
2 0
0 1
1
1
1
2 0
2
2
0 2
1 2
2 2
2 1 0 0
0 0
1 0
1 0
0 2
1 0
0 1 0 2
2 0
1 2
1
2
1
0 1
2
1
0
2
0
2
0 2
0
2
3 0 17 7 7 07 7 7 17 7 07 7; 7 17 7 07 7 7 15 0 3 0 27 7 7 07 7 7 27 7 07 7: 7 27 7 07 7 7 25 0
TH ¼ At Thom where A is the ðjEj jEjÞ-matrix given by ARx;Ry ¼
jR xj zðRx; RyÞ: jRyj
328
MARCUS GREFERATH
The matrix A is invertible, and its inverse B is given by BRx;Ry ¼
jRxj mðRx; RyÞ: jR yj
Here z and m are the respective zeta and Mo. bius functions on the poset E. Proof. have
By Remark 2.4 and appropriate homomorphism arguments, we 1 X 1 X whom ðrxzÞ ¼ whom ðyzÞ jRj r2R jRxj y2Rx X jR yj 1 X zðRy; RxÞwhom ðyzÞ; ¼ whom ðyzÞjR yj ¼ jRxj Ry Rx jRxj Ry2E
wH ðxzÞ ¼
which shows the first part of our claim. For the second part we verify ðBAÞRx;Rz ¼
X
BRx;Ry ARy;Rz
Ry2E
X jRxj jR yj zðRy; RzÞ mðRx; RyÞ jR yj jRzj Ry2E jRxj X ¼ mðRx; RyÞ ¼ dðRx; RzÞ: jRzj Ry Rz ¼
&
Let us now show the main result which assumes the underlying ring to be a finite Frobenius ring. Theorem 3.3. If R is a finite Frobenius ring and M ¼ Rk , then Thom and hence also TH is a ðsquareÞ invertible matrix. Proof. We will introduce elementary row operations on Thom which produce an invertible ðjE * j jE * jÞ-matrix. This will in particular reveal that jEj jE * j, which by symmetry forces jEj ¼ jE * j. For arbitrary D R Rk first consider the expression ðfyR ðDÞÞyR2E * where fyR ðDÞ :¼
1 X whom ðxyÞjR xj: jDj Rx D
Being of length jE * j the vector ðfyR ðDÞÞyR2E * is a weighted sum over all rows of Thom , where the row indices range over the nonzero cyclic submodules
MACWILLIAMS’ EQUIVALENCE THEOREM
329
of D. Simple homomorphism arguments show that 1 X 1 X fyR ðDÞ ¼ whom ðx; yÞjR xj ¼ whom ðxyÞ jDj Rx D jDj x2D 1 X ¼ whom ðzÞ ¼ 1 dðDy; 0Þ; jDyj z2Dy as a consequence of Remark 2.3. Now, if D ¼ Rk , we have fyR ðDÞ ¼ 1 for all yR 2 E * . If D ¼ ðzRÞ? for some zR 2 E * , then fyR ðDÞ ¼ 1 zðyR; zRÞ: Combining this it is clear that fyR ðRk Þ fyR ððzRÞ? Þ ¼ zðyR; zRÞ; which means that we have expressed the zeta function of the partial order on E * by row combinations of Thom . Our claim now follows from the fact that the matrix representing z is invertible. & 4.
THE EQUIVALENCE THEOREM
Let a finite Frobenius ring R and an R-linear code C of length n be given. Assume that C is k-generated; i.e., there exists a ðk nÞ-generator matrix G for C. Corresponding to what we have discussed in the previous sections, we define E * :¼ fyR j 0=y 2 Rk g and consider the rational vector r :¼ ðryR ÞyR2E * where ryR :¼ jfc j c is a column of G; with cR ¼ yRgj: Clearly, r can be considered as a multiset on E * counting all nonzero columns of G up to associates. Observation 4.1 For all x 2 Rk there holds and ðThom rÞRx ¼ whom ðxGÞ: ðTH rÞRx ¼ wH ðxGÞ Proof Let w be one of wH or whom , and denote by T the corresponding orthogonality matrix on Rk . For all Rx 2 E we easily verify X ðTrÞRx ¼ wðxyÞ#fc j c is a column of G; with cR ¼ yRg yR2E *
¼
X
wðxcÞ ¼ wðxGÞ:
c2Rk c column of G
This prove our claim.
&
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MARCUS GREFERATH
Theorem 4.2. For a finite Frobenius ring R, a left R-linear code C of length n, and an R-linear mapping C !j Rn , the following are equivalent: (a) j is a Hamming ðresp. homogeneousÞ isometry. (b) j is restriction of a monomial transformation of R Rn . Proof. The necessity of (a) for (b) is clear. Therefore assume (a) and let G be a ðk nÞ-generator matrix for C. Furthermore, let T be the respective orthogonality matrix for the underlying module M :¼ Rk . It is clear that G0 :¼ Gj (defined row-wise) is a ðk nÞ-generator matrix for C 0 :¼ Cj. For the multisets r and r0 induced by G and G0 we obtain the following: as G0 ¼ Gj we have wðxGÞ ¼ wðxG0 Þ for all x 2 C. Therefore Tr ¼ Tr0 by Observation 4.1, and hence r ¼ r0 because of Theorem 3.3. Consequently the columns of G and G0 are the same up to permutation and (right) associates, i.e., up to a monomial transformation of Rn . &
ACKNOWLEDGMENT The author thanks Sergio Lop!ez-Permouth for the opportunity to visit the Mathematics Department of Ohio University, where the results of the present paper were developed.
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