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APPLICATIONS OF FINITE FROBENIUS RINGS TO THE FOUNDATIONS OF ALGEBRAIC CODING THEORY JAY A. WOOD Abstract. This article addresses some foundational issues that arise in <strong>the</strong> study <strong>of</strong> linear codes defined over finite rings. Linear coding <strong>the</strong>ory is particularly well-behaved over finite Frobenius rings. This follows from <strong>the</strong> fact that <strong>the</strong> character module <strong>of</strong> a finite ring is free if and only if <strong>the</strong> ring is Frobenius. Key Words: Frobenius ring, generating character, linear code, extension <strong>the</strong>orem, MacWilliams identities. 2010 Ma<strong>the</strong>matics Subject Classification: Primary 16P10, 94B05; Secondary 16D50, 16L60. 1. Introduction At <strong>the</strong> center <strong>of</strong> coding <strong>the</strong>ory lies a very practical problem: how to ensure <strong>the</strong> integrity <strong>of</strong> a message being transmitted over a noisy channel? Even children are aware <strong>of</strong> this problem: <strong>the</strong> game <strong>of</strong> “telephone” has one child whisper a sentence to a second child, who in turn whispers it to a third child, and <strong>the</strong> whispering continues. The last child says <strong>the</strong> sentence out loud. Usually <strong>the</strong> children burst out laughing, because <strong>the</strong> final sentence bears little resemblance to <strong>the</strong> original. Using electronic devices, messages are transmitted over many different noisy channels: copper wires, fiber optic cables, saving to storage devices, and radio, cell phone, and deep-space communications. In all cases, it is desirable that <strong>the</strong> message being received is <strong>the</strong> same as <strong>the</strong> message being sent. The standard approach to error-correction is to incorporate redundancy in a cleverly designed way (encoding), so that transmission errors can be efficiently detected and corrected (decoding). Ma<strong>the</strong>matics has played an essential role in coding <strong>the</strong>ory, with <strong>the</strong> seminal work <strong>of</strong> Claude Shannon [27] leading <strong>the</strong> way. Many constructions <strong>of</strong> encoding and decoding schemes make strong use <strong>of</strong> algebra and combinatorics, with linear algebra over finite fields <strong>of</strong>ten playing a prominent part. The rich interplay <strong>of</strong> ideas from multiple areas has led to discoveries that are <strong>of</strong> independent ma<strong>the</strong>matical interest. This article addresses some <strong>of</strong> <strong>the</strong> topics that lie at <strong>the</strong> ma<strong>the</strong>matical foundations <strong>of</strong> algebraic coding <strong>the</strong>ory, specifically topics related to linear codes defined over finite rings. This article is not an encyclopedic survey; <strong>the</strong> ma<strong>the</strong>matical questions addressed are ones in which <strong>the</strong> author has been actively involved and are ones that apply to broad classes <strong>of</strong> finite rings, not just to specific examples. Prepared for <strong>the</strong> <strong>44th</strong> <strong>Symposium</strong> on Rings and Representation Theory Japan, 2011. Supported in part by a sabbatical leave from Western Michigan University. This paper is in final form and no version <strong>of</strong> it will be submitted for publication elsewhere. –223–
The topics covered are ring-<strong>the</strong>oretic analogs <strong>of</strong> results that go back to one <strong>of</strong> <strong>the</strong> early leaders <strong>of</strong> <strong>the</strong> field, Florence Jessie MacWilliams (1917–1990). MacWilliams worked for many years at Bell Labs, and she received her doctorate from Harvard University in 1962, under <strong>the</strong> direction <strong>of</strong> Andrew Gleason [22]. She is <strong>the</strong> co-author, with Neil Sloane, <strong>of</strong> <strong>the</strong> most famous textbook on coding <strong>the</strong>ory [23]. Two <strong>of</strong> <strong>the</strong> topics discussed in this article are found in <strong>the</strong> doctoral dissertation <strong>of</strong> MacWilliams [22]. One topic is <strong>the</strong> famous MacWilliams identities, which relate <strong>the</strong> Hamming weight enumerator <strong>of</strong> a linear code to that <strong>of</strong> its dual code. The MacWilliams identities have wide application, especially in <strong>the</strong> study <strong>of</strong> self-dual codes (linear codes that equal <strong>the</strong>ir dual code). The MacWilliams identities are discussed in Section 4, and some interesting aspects <strong>of</strong> self-dual codes due originally to Gleason are discussed in Section 6. The o<strong>the</strong>r topic to be discussed, also found in MacWilliams’s dissertation, is <strong>the</strong> MacWilliams extension <strong>the</strong>orem. This <strong>the</strong>orem is not as well known as <strong>the</strong> MacWilliams identities, but it underlies <strong>the</strong> notion <strong>of</strong> equivalence <strong>of</strong> linear codes. It is easy to show that a monomial transformation defines an isomorphism between linear codes that preserves <strong>the</strong> Hamming weight. What is not so obvious is <strong>the</strong> converse: whe<strong>the</strong>r every isomorphism between linear codes that preserves <strong>the</strong> Hamming weight must extend to a monomial transformation. MacWilliams proves that this is indeed <strong>the</strong> case over finite fields. The MacWilliams extension <strong>the</strong>orem is a coding-<strong>the</strong>oretic analog <strong>of</strong> <strong>the</strong> extension <strong>the</strong>orems for isometries <strong>of</strong> bilinear forms and quadratic forms due to Witt [30] and Arf [1]. This article describes, in large part, how <strong>the</strong>se two results, <strong>the</strong> MacWilliams identities and <strong>the</strong> MacWilliams extension <strong>the</strong>orem, generalize to linear codes defined over finite rings. The punch line is that both <strong>the</strong>orems are valid for linear codes defined over finite Frobenius rings. Moreover, Frobenius rings are <strong>the</strong> largest class <strong>of</strong> finite rings over which <strong>the</strong> extension <strong>the</strong>orem is valid. Why finite Frobenius rings? Over finite fields, both <strong>the</strong> MacWilliams identities and <strong>the</strong> MacWilliams extension <strong>the</strong>orem have pro<strong>of</strong>s that make use <strong>of</strong> character <strong>the</strong>ory. In particular, finite fields F have <strong>the</strong> simple, but crucial, properties that <strong>the</strong>ir characters ̂F form a vector space over F and ̂F ∼ = F as vector spaces. The same pro<strong>of</strong>s will work over a finite ring R, provided R has <strong>the</strong> same crucial property that ̂R ∼ = R as onesided modules. It turns out that finite Frobenius rings are exactly characterized by this property ([14, Theorem 1] and, independently, [31, Theorem 3.10]). The character <strong>the</strong>ory <strong>of</strong> finite Frobenius rings is discussed in Section 2, and <strong>the</strong> extension <strong>the</strong>orem is discussed in Section 5. Some standard terminology from algebraic coding <strong>the</strong>ory is discussed in Section 3. While much <strong>of</strong> this article is drawn from earlier works, especially [31] and [33], some <strong>of</strong> <strong>the</strong> treatment <strong>of</strong> generating characters for Frobenius rings in Section 2 has not appeared before. The new results are marked with a dagger (†). Acknowledgments. I thank <strong>the</strong> organizers <strong>of</strong> <strong>the</strong> <strong>44th</strong> <strong>Symposium</strong> on Rings and Representation Theory Japan, 2011, especially Pr<strong>of</strong>essor Kunio Yamagata, for inviting me to address <strong>the</strong> symposium and prepare this article, and for <strong>the</strong>ir generous support. I thank Pr<strong>of</strong>essor Yun Fan for suggesting subsection 2.4 and Steven T. Dougherty for bringing <strong>the</strong> problem <strong>of</strong> <strong>the</strong> form <strong>of</strong> a generating character to my attention (answered by Corollary 15). I also thank M. Klemm, H. L. Claasen, and R. W. Goldbach for <strong>the</strong>ir early work –224–
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Proceedings <stron
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Organizing Committee of</st
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Polycyclic codes and sequential cod
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Preface The 44th <
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8:40-9:30 Dan ZachariaSyracuse Univ
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The 44th S
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Tuesday September 27 8:40-9:30 Atsu
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Conversely, let (T , F) be a stable
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Then T = gen(X) and X is Ext-projec
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DIMENSIONS OF DERIVED CATEGORIES TA
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Notation 4. (1) Let A be an abelian
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of finitely genera
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is the map given b
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(2) Let R be a commutative ring whi
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QUIVER PRESENTATIONS OF GROTHENDIEC
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Proposition 3. Let C be a category,
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Example 11. Let Q be the</s
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and Gr(X ′ ) is given by
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particular, the de
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Proposition 1 and 2 leave us with d
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4. Examples Example 6. Let M = M(A
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EXAMPLE OF CATEGORIFICATION OF A CL
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The aim of <strong
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X ∈ mod Π Q S 1 S 2 S 3 1 ❁
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The previous proposition has a dual
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Computing inductively all t
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Example 23. We have ⎛ ⎞ ⎛ ⎞
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classification of
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quiver Q over K, and let D b (H) <s
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Then Q ∈ Q 17 . Suppose char K
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Moreover the Hochs
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DERIVED AUTOEQUIVALENCES AND BRAID
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3. Periodic Twists We now describe
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We now specialise to the</s
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and τ − n := Ext n Λ(DΛ, −)
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where r v ij := a i a j −a j a i
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ON A DEGENERATION PROBLEM FOR COHEN
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We make several othe</stron
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Let A be a commutative Gorenstein r
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3. Extended orders In the</
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WEAK GORENSTEIN DIMENSION FOR MODUL
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1.2. Introduction. The notion <stro
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e 2 A z 1 −→ X → 0 in mod-A,
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5. Gorenstein algebras In this sect
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ON Ω-PERFECT MODULES AND SEQUENCES
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when R is a Nakayama algebra <stron
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says that components of</st
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then we obtain a c
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Proposition 16. Let C s be a stable
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3.1. ZD ∞ -components. We assume
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Theorem 26. Let R be a local artini
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where each f i is an irreducible ep
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QUANTUM UNIPOTENT SUBGROUP AND DUAL
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{F i } i∈I and q-Serre relations
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Theorem 8. (1)Let U − q (w) → U
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E-mail address: ykimura@kurims.kyot
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Indeed, (1, 2, 3, 4) {1,2} −−
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By comparing this the</stro
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(a) (b) Proposition 17. Let G be a
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POLYCYCLIC CODES AND SEQUENTIAL COD
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⎛ ⎞ 0 0 c 0 1 c t D c = ⎜ 1
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References [1] D. Boucher and P. So
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2. Dimension of tr
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APR TILTING MODULES AND QUIVER MUTA
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(1) Q ′ is a quiver obtained from
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1 arrows of ˜µ L
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[11] S. Fomin, A. Zelevinsky, Clust
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Theorem. Assume r is a common prime
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References [1] Apostol, T. M., The
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e(n, s) n = 6 7 8 s = 1 36 63 120 2
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Now we will show that the</
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is equal to ( n 2s−1 ); hence we
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THE NOETHERIAN PROPERTIES OF THE RI
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Proposition 4 (Paper III, Propositi
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HOCHSCHILD COHOMOLOGY OF QUIVER ALG
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(4) In [10], Green, Snashall and So
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4. Quiver algebras defined by two c
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[6] L. Evens, The cohomology ring <
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Then there is an i
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• ( ˜F , ˜m) = ({ (1, 3), (2, 3
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Corollary 7 ([16, Theorem 3.4]). Th
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We have the direct
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We say a CW complex is regular, if
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SHARP BOUNDS FOR HILBERT COEFFICIEN
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Let us briefly note how this paper
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whence the set Λ
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Proof of</
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We have Q·H j m(A/(a 1 , a 2 , ·
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Let B = A/U(a d−1 ). We t
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• quiver Γ = (I, Ω) I Ω
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B = (B τ ) relations ρ 1 , · ·
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Definition 1. double quiver ˜Γ p
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Page 197 and 198: B ∈ Λ(d) ε ∗ i (B) := dim C
Page 199 and 200: I-graded vector space V (d) (B τ )
Page 201 and 202: A ∗(i) l (a) = ∑n+1 t=l+1 (a i,
Page 203 and 204: wt(a) = (−d 1 , −d 2 , −d 3 )
Page 205 and 206: Ω Q Ω ( B Ω ; wt, ε i , ϕ
Page 207 and 208: “Λ(d) G(d)-” Definition 25.
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Page 211 and 212: MATRIX FACTORIZATIONS, ORBIFOLD CUR
Page 213 and 214: ( ∂f Jac(f t t ) := C[x 1 , . . .
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Page 221 and 222: (f, G f ) Dolgachev A Gf f Berg
Page 223 and 224: ON A GENERALIZATION OF COSTABLE TOR
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Page 231 and 232: GRADED FROBENIUS ALGEBRAS AND QUANT
Page 233 and 234: Definition 2. A graded algebra A is
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Page 237: References [1] X. W. Chen, Graded s
Page 241 and 242: (2.3) soc( R R) ∼ = k⊕ s i T i
Page 243 and 244: principal ideal Rr ⊂ ker ϖ. Thus
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(Proof) ( ) p 0 0
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we have a sequence of</stro
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R : CM(R⊗ k V ) → CM(R) defined
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P = P ′ t by t is a projective R
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SUBCATEGORIES OF EXTENSION MODULES
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Lemma 6. Let S 1 and S 2 be Serre s
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In the first row <
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