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Constructions of MDS-Convolutional Codes Ring ‘H“, with R—Nk Q@Haak

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Constructions of MDS-Convolutional Codes Ring ‘H“, with R—Nk Q@Haak IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 5, JULY 2001 2045 Constructions of MDS-Convolutional Codes ring h, with rnk q@hAak. For the purpose of this correspon- dence, we define the rate kan convolutional code generated by q@hA Roxana Smarandache, Student Member, IEEE, as the set Heide Gluesing-Luerssen, and n k Joachim Rosenthal, Senior Member, IEEE g a fu@hAq@hA P @hAju@hA P @hAg and say that q@hA is a generator matrix for the convolutional code g q@hA qH@hA Abstract—Maximum-distance separable (MDS) convolutional codes are . If the generator matrices and both generate the same characterized through the property that the free distance attains the gener- convolutional code g then there exists a k k invertible matrix @hA alized Singleton bound. The existence of MDS convolutional codes was es- with qH@hAa @hAq@hA and we say q@hA and qH@hA are equiva- tablished by two of the authors by using methods from algebraic geometry. lent encoders. This correspondence provides an elementary construction of MDS convo- lutional codes for each rate and each degree . The construction is Because of this, we can assume without loss of generality that the based on a well-known connection between quasi-cyclic codes and convo- code g is presented by a minimal basic encoder q@hA. For this, let lutional codes. #i be the ith-row degree of q@hA, i.e., #i amxj deg gij @hA. In the # Index Terms—Convolutional codes, generalized Singleton bound, max- literature [7], the indexes i are also called the constraint length for the imum-distance separable (MDS) convolutional codes. ith input of the matrix q@hA. Then one defines the following. Definition 1.1: A polynomial generator matrix q@hA is called basic k # I. INTRODUCTION if it has a polynomial right inverse. It is called minimal if iaI i attains the minimal value among all generator matrices of g. The free distance of a rate kan convolutional code of degree is always upper-bounded by the generalized Singleton bound A basic generator matrix is automatically noncatastrophic, this means finite-weight codewords can only be produced from fi- dfree @n kA@ak CIAC CI (1.1) nite-weight messages. If q@hA is a minimal basic encoder one defines k the degree [8] of g as the number Xa iaI #i. In the literature, see [1]. We will provide an alternative proof of this result in the next the degree is sometimes also called the total memory [9] or the section. If aH, i.e., in the case of block codes, (1.1) simply reduces overall constraint length [7] or the complexity [10] of the minimal to the well-known Singleton bound basic generator matrix q@hA, a number dependent only on g. Among all these equivalent expressions we like the term degree best since d n k CI free (1.2) it relates naturally to equal objects appearing in systems theory and algebraic geometry. The following remarks explain this notion. cf. [2, Ch. 1, Theorem 11]. The authors of [1] showed the existence of rate kan convolutional codes of degree whose free distance was Remark 1.2: It has been shown by Forney [11] that the set equal to the generalized Singleton bound (1.1) and they called such f#IY FFFY#kg of row degrees is the same for all minimal basic codes maximum-distance separable (MDS) convolutional codes. The encoders of g. Because of this reason, McEliece [8] calls these indexes existence was established in [1] by techniques from algebraic geometry the Forney indexes of the code g. These indexes are also the same as without giving an explicit construction. This correspondence is based the Kronecker indexes of the row module on ideas from Justesen [3] and it provides an explicit construction of w a fu@hAq@hA P nhju@hA P khg MDS convolutional codes for each rate kan and each degree . The construction itself uses a well-known connection between quasi-cyclic when q@hA is a basic encoder. The Pontryagin dual of w defines a codes and convolutional codes which has been worked out by several linear time-invariant behavior in the sense of Willems [12], [13], i.e., a authors [3]–[6]. linear system. Under this identification, the Kronecker indexes of w The correspondence is structured as follows. In the remainder of this correspond to the observability indexes of the linear system [14]. The section we introduce the basic notions which will be needed throughout sum of the observability indexes is equal to the McMillan degree of the correspondence. In Section II, we give a new derivation of the gen- the system. Finally, w defines in a natural way a quotient sheaf [15] eralized Singleton bound (1.1). The main new results will be given in over the projective line and, in this context, one refers to the indexes Section III. f#IY FFFY#kg as the Grothendieck indexes of the quotient sheaf and Let be a finite field, h the polynomial ring, and @hA the field k a iaI #i as the degree of the quotient sheaf. of rational functions. Let q@hA be a k n matrix over the polynomial We feel that the degree is the single most important code parameter on the side of the transmission rate kan. In the sequel, we will adopt Manuscript received September 15, 1999; revised November 30, 2000. This the notation used by McEliece [8, p. 1082] and denote by @nY kY A a work was supported in part by NSF under Grants DMS-96-10389 and DMS-00- rate kan convolutional code of degree . 72383. The work of R. Smarandache was supported by a fellowship from the n Center of Applied Mathematics at the University of Notre Dame. J. Rosenthal For any n-component vector v P , we define its weight wt @vA as was carrying out this work while he was a Guest Professor at EPFL in Switzer- the number of all its nonzero components. The weight wt @v@hAA of a land. The material in this correspondence was presented in part at the 2000 IEEE vector v@hA P n@hA is then the sum of the weights of all its n-co- International Symposium on Information Theory, Sorrento, Italy, June 25–30, efficients. Finally, we define the free distance of the convolutional code 2000. n R. Smarandache and J. Rosenthal are with the Department of Mathematics, g& @hA through University of Notre Dame, Notre Dame, IN 46556 USA (e-mail: Smaran- [email protected]; [email protected]). dfree aminfwt @v@hAAjv@hA PgYv@hA TaHgX (1.3) H. Gluesing-Luerssen is with the Department of Mathematics, Universität Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany (e-mail: gluesing@ It is an easy but crucial observation that in case we are given a basic mathematik.uni-oldenburg.de). encoder q@hA the free distance can also be obtained as Communicated by R. M. Roth, Associate Editor for Coding Theory. Publisher Item Identifier S 0018-9448(01)04751-4. dfree a minfwt @v@hAAjv@hA PwYv@hA TaHgX 0018–9448/01$10.00 © 2001 IEEE 2046 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 47, NO. 5, JULY 2001 This follows simply from the fact that, if q@hA has a polynomial right Once the row degrees #IY FFFY#k of the minimal basic encoder inverse, a nonpolynomial message u@hA would result in a nonpolyno- q@hA are specified one has a natural upper bound on the free distance mial codeword v@hA, which, of course, has infinite weight. of a convolutional code. The following result was derived in [1]. In the sequel, we wish to link the free distance to two types of dis- Theorem 2.1: Let g be a rate kan convolutional code generated by tances known from the literature. Following the approach in [16], [7] r a minimal-basic encoding matrix q@hA. Let #IY FFFY#k be the row we shall define the column distances dj and the row distances dj .In degrees of q@hA and # a minf#IY FFFY#kg denote the value of the order to do so let us suppose smallest row degree. Finally, let be the number of indexes #i among q@hAaq C q h C q hP C C q h# H I P # the indexes #IY FFFY#k having the value #. Then the free distance must is an encoder with row degrees #I #k. Denote by (1.4) the satisfy semi-infinite sliding generator matrix, as shown at the bottom of the dfree n@# CIA CIX (2.1) page. Then the convolutional code can be defined as k g a f@uHYuIY FFFYu Y FFFA qjuj P Y for j aHY IY FFFgX The proof given in [1] was based on the polynomial generator matrix q@hA. In the sequel, we provide a proof by means of the sliding matrix j d Then the th-order column distance j is defined as the minimum of q introduced in (1.4). v Xa @v YvY FFFYvA the weights of the truncated codewords HYj H I j re- Proof: Without loss of generality, we may assume sulting from an information sequence uHYj Xa @uHYuIY FFFYuj A with # a # # X uH TaH. Precisely, if qj denotes the k@jCIAn@jCIAupper-left sub- I k matrix of the semi-infinite matrix q, then dj aminu TaH wt@uHYj Let q be the infinite sliding generator matrix associated to q@hA as qj AX The quantity d# is called the minimum distance of the code and in (1.4).
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