CONVOLUTIONAL CODES AND IRREDUCIBLE IDEALS

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1 R 78 Philips Res. Repts 27, 27-27, 72 CONVOLUTIONAL CODES AND IRREDUCIBLE IDEALS by Ph. PIRET Abstract.RS-like and BCH-like convolutional codes are constructed, which seem to have good free-distance properties. Some BCH-like codes are optimal for the Plotkin bound. The RS-like codes are extensible by pseudo random encoding to obtain long asymptotically good convolutional codes.. Introduction The construction of "good" block or convolutional codes is not an easy matter. Short BCH.2) or quadratic residue ) block codes are good enough and sometimes optimal. On the other hand, Justesen 26) recently constructed infinite families of block codes with a fixed rate kin, for which djn remains larger than a computable strictly positive number only depending on the chosen rate, when n grows to infinity. One of the steps in his construction uses the Reed-Solomon.8) codes. Two different points of view can be adopted to construct convolutional codes. One can try to obtain codes with an algebraically describable decoding algorithm.6.-2.). A different possibility is to search for codes with a large free distance, which are well suited for sequential decoding ). Except for the last, these constructions are algorithmic and the structure of the codes has no simple algebraic description. On the other hand, these codes only have one information symbol per word. The constructions we shall describe in this paper make use of block codes and of the bounds obtained for their minimum distance, and in this sense, they present some similarities with some other results 0-2.). However, these latter results are largely generalized and more systematically obtained. On the other hand, our main interest is in the free' distance d f. We shall obtain algebraically describable convolutional codes with a lower bound on d f by use of arguments developed by other authors ). These arguments are either algebraic, or based on a sequential procedure. We shall thus obtain RS-like and BCH-like convolutional codes. Our RS-like convolutional codes can be extended by the pseudo random encoding mode of Justesen to obtain families of convolutional codes of fixed rate and constraint length ', for which n grows to infinity while dflvn remains larger than a computable strictly positive number.

2 28 Ph. PIRET 2. A convenient representation for the convolutional codes Let q be a power of 2, and consider a linear convolutional code of length n, rate kin, and constraint length ")I, on GF(q). A generator matrix for this code can be represented as where all G) are (k,n) matrices, with elements in GF(q). A generator matrix can also be represented as () v -e L (2) and the use of both representations is well known. We now suppose that the block codes generated by all G) are cyclic. Since a cyclic code of length nis the direct sum of its irreducible ideals in the algebra of polynomials modulo (xn_ ), we can represent each G l as follows: Gl.) G2,) G)= () Gm,) where all GIJ are generator matrices of different irreducible cyclic codes. Consider now 0:, a primitive nth root of unity; 0: is in GF(q) or in some of its proper extensions. We denote by m,(x) the minimal polynomial of 0:' with coefficients in GF(q). It is a factor of (xn_ ), and a code defined by any GIJ can also be seen as the irreducible ideal in the algebra of polynomials modulo (xn_ ), generated by xn_ gr(x) = mr(x) (4) for some convenient r. Since m,(x) is the minimal polynomial not only of 0:', but also of all 0: with i = rs' modulo n, for s =,2,..., the index r is univocally fixed by choosing the smallest of these i after reduction modulo n.. In the following, we shall denote all the Gij only by the index r of the corresponding m,(x). Moreover, all Gij with a fixed i will correspond to irreducible ideals of the same dimension. With these conventions the generator matrix () now becomes a rectangular array of integers. This representation is well adopted for our purpose. An

3 CONVOLUTIONAL CODES AND IRREDUCIBLE IDEALS 2 example is now given. Consider a code of length on GF(2); ()(is a root of XS + X 2 +, and the other ml(x) can be found in the book of Peterso? 4): m (X) =Xs +X2 +, m(x) = XS + X4 + X + X2 +, ms(x) = XS + X4 + X 2 + X +, m7(x) = XS + X + X 2 + X +, m ll (X)=XS+X4+X +X+, m S (X) = Xs + X +, mo(x) = X +. () A (, 0) convolutional code with = can have the following matrix G: G = [ J 7 ' (6) In this matrix each number r refers to a generator matrix of a (, ) cyclic code, whose generator matrix is (X _ l)fm r (X). From any matrix G we obtain easily a semi-infinite matrix r such that all sequences of the code are in the "row space" of r. For the above example we obtain 7 ~ 7 r= (7) 7 7. A sequence of the code can thus be partially characterized by a set of rows of I', referring to the non-zero components of its information symbols.. An easy way to check the non-catastrophic character of a generator matrix It is well known that a code exhibits catastrophic error propagation, if and only if there is an information sequence of infinite weight that gives a fini teweight encoded sequence 4.20). We now investigate the structure of such a finite weight, infinite sequence for the codes defined in sec. 2.

4 260 Ph. PlRET Any vector expressed in the basis of the irreducible ideals can be zero if and only if all its components are zero, so that any number of G that appears an infinite number of times in a catastrophic sequence, must appear strictly more than once in each column where it appears, and this at least after some wth column, where w is finite. If there is no finite w for which this is verified, the weight of the sequence is necessarily infinite. Let now y be the set of rows of G that can appear an infinite number of times in a catastrophic sequence. We first suppose that all rows of G are in y. Since each number that appears in these rows must appear at least twice in y, we can subtract from y all rows containing numbers that appear only once in G. The same process can be iterated until all elements of the remaining rows appear at least twice in the set of these rows. If y does not become empty the problem remains open. But if y becomes the empty set, the code surely presents no catastrophic error propagation. As an example we consider a code of length 7 on GF(2 ). All mr(x) have degree. This code has the following generator matrix: 2 G= o The elements and 0 appear only once in this matrix, so that in a first step y ~n is reduced to [! Now, 2 and 6 appear once, and y is reduced to [ 4]. This last row can also be deleted since all its components appear once. The set y becomes empty and the code is surely not catastrophic. 4. The low-rate RS-Iike convolutional codes 4.. Primitive convolutional codes We now apply the concepts of sec. 2 to the case where a is a primitive element of GF(q), (q = 2"', n = 2'" - ). All m,(x) have degree, and the generators for the irreducible ideals are XZm-l_ gr(x) = X r ' r = 0,,..., 2"'- 2. -a All matrices Gij thus have one row and (2'" - ) columns and will be repre-.sented by the corresponding r (8) () /

5 CONVOLUTIONAL CODES AND IRREDUCIBLE IDEALS 26 Let us consider the low-rate RS-like convolutional code of length (2'" - ) on GF(2"') defined by the following matrix G: G= o k j j ~I (,-) kl k + I. : ('-r+ IJ' (0) k-l 2k-l (j+l)k-l vk-l where v k is at most 2"'. For v k = 2 m, we have v k- = 0, since We now investigate the free distance of the code defined by G. Theorem 4.. The code defined by (0) is not catastrophic and has a free distance d f satisfying d f ~ v (2 m _ Proof It is first easily proved by use of sec. that this code is not catastrophic. Only finite sequences must thus be considered to determine the free distance- Consider now an information sequence of length u, i.e. a sequence of informa. tion k-tuples that is identically zero before the first and after the uth of these. The first and the uth information k-tuples are not zero. We first suppose that ~ u~ v-. For all i ~ u-, the ith word in the encoded sequence is a nonzero word of a cyclic code whose generators are the gr(x) with 0 ~ r ~ i k- Its minimal weight 6) is thus at least (2 m - ik). Similarly, the (u + v- i)th word is generated by the gr(x), with (v- i)k ~ r ~ vk-, so that its weight is also at least (2'"- ik) 6) and the intermediate words from the uth to the vth have a weight ~ (2'"- u k). The minimal weight of this sequence is thus lower-bounded by u-i 2 l: (2'"_ ik) + (v- u + )(2 m _ uk) = (v- ) 2'" + u (2'"_ v k). () = In the range of u and v considered, this expression is minimal for u =, where its value is v (2 m - k). We now suppose that u ~ v. As in the first case, both the weights of the ith and of the (u+ v- i)th word, ( ~ i ~ v- ), are at least (2'"- i k), so that the total weight of the sequence is at least k). v-i 2l: (2 m _ ik) = 2 (v- )2 m - v (v- )k. = (2)

6 262 Ph. PIRET It is then easily checked that this last expression is greater than or equal to v (2 m - k), or equivalently that for all positive integers v, from 2 to 2'"jk. Q.E.D Non-primitive convolutional codes (v - 2) 2 m - v (v - 2) k ~ 0 () The above construction is easily extended to all odd lengths n. Let a be a primitive nth root of unity and GF(2 ) be the smallest field that contains a. The numbers r, of a generator matrix, now refer to the irreducible ideals generated by the polynomials gr(x), defined on the field GF(2 m ), as follows: xn-l gr(x) = ---, r = 0,,..., n -. X - ar \ The code is now also defined by the matrix G, (eq. (0)), where v k is at most n +. If v k = n + I, v k - is zero since an = (l0 =. Theorem.2. The code we just defined is not catastrophic and has a free distance that is at least v (n + - k). Proof It follows from straightforward generalizations of the preceding one, and it will be omitted. As an example, consider a code of length 2 on GF(2048) generated by a, a primitive 2th root of unity. Such a code with 6 information symbols per word, and v = 4, can be specified by the following matrix: G= Its free distance is at least 4. ( ) = Other distance properties of the RS-like convolutional codes Besides the free distance d f and the minimum distance d mn, the following distances can be defined for a convolutional code 2): ' (a) dei) is the column distance of order i. It is an increasing function defined as being the minimal weight of all sequences of the code, beginning by a non-zero word and limited to their (i + ) first words. In particular, d (v - ) is the minimum distance of the code. All dei) are upper-bounded by d f (4)

7 CONVOLUTIONAL CODES AND IRREDUCIBLE IDEALS 26 (b) rei) is the row distance of order i.lt is a decreasing function defined as being the minimal weight of the code sequences beginning with a non-zero word, and corresponding to information sequences that are zero after their (i + I)th k-tuple. All rei) are lower-bounded by df. The preceding paragraphs lead us to the following remarks. (RI) The lower bound obtained for rei) is already df for reo); r(i) thus seems to be a weakly decreasing function for the RS-like convolutional codes. (R2) Denoting by dei) a lower bound on dei), obtained as in the proof of theorem 4., we may write d(i) ~ d(i- )+ (n + )- (i + )k. lt is thus a fairly increasing function. In particular, one has () 2 (n + ) - k (v + ) d(v- ) ~ v. 2 (6) Both remarks suggest that the RS-like convolutional codes are "good" codes.. High-rate R,S-like convolutional codes The codes defined in sec. 4 are low-rate codes whose free distance is Iower- bounded by df~v(n-k+ I). (7) These constructions give codes with k v-i ~ n and are not applicable to rates greater than t. Consider now a code defined by ex,a primitive nth root of unity, that has the following generator matrix: o G= 2 2 (8) k-l k If k ~ n-, it is easy to check that this code is not catastrophic and has a free distance lower-bounded by (7) for v = 2. Consider now a code with v = and k ~ n- I, defined by G= 0 L () k-i k\ k-i

8 264 Ph. PIRET This code is clearly non-catastrophic, and it is not difficult to see that each finite encoded sequence of the code contains at least three non-zero words. If k ;;;::: 2, it is easily shown that (7) remains valuable. The complete proof is left to the reader. ; 6. Free distance of binary convolutional codes The free distance of the binary convolutional codes as defined in sec, can be computed with the aid of two different arguments. The first is algebraic and uses known bounds on the minimum distance of block codes. The second is sequential and applies ideas of Forney 2), to a stack-like algorithm similar to that of Jelinek 8) and Zigangirov 7). 6.. Algebraic arguments These are more easily described with the aid of an example. Consider the (, 0) code whose matrix G was given hereabove (6). We shall first examine the structure of the cyclic block codes of length, and construct therefore table I as follows. We choose an exthat is a primitive th root of unity, and from this exwe construct the corresponding set of polynomials mr(x) as in sec. 2. We then range the irredundant set of indices r, except 0, in increasing order. This is the first row of table I: (,,, 7,, ), where each r represents the class of the numbers r 2' modulo, for s =, 2,, 4,. Consider now ex a different primitive th root of unity, chosen on such a way that, the exponent of ex,is not in the class of.the substitution ex-+ ex results in a permutation of the r in the first row of table I, i.e. the polynomial mi (X) corresponding to ex is in fact the polynomial m(x) corresponding to ex.similarly m~(x) corresponding to ex is ms(x) corresponding to ex,and so on. The second row of table I will contain the permutation of the indices r, induced by the substitution ex-+ ex In the same way the four last rows will contain the permutations induced by the substitutions ex-+ ex r for r =, 7,,. In general, all rows corresponding to indices r that are not relatively prime with n must be deleted, and the table is square only if n is a prime. TABLE I

9 CONVOLUTIONAL CODES AND IRREDUCIBLE IDEALS 26 The theory of the BCH codes shows us that a word generated by some given set of g,(x), each with a non-zero coefficient, has a minimum weight that can be lower-bounded as follows. Suppose first that all r are different. We then first search the row in the above table where the set of these indices rappears the mos.-to the right. Suppose that in this row no considered index appears in the (w - ) first columns. The minimum weight of the word is lower-bounded byadding to the leader of the wth column. For example, a word of length, generated by gl(x) and gl(x) is characterized by the set (, ). This set appears in the last columns of the rd row of table. The minimum weight of the word is thus at least 8. Suppose now that the set of r contains multiple indices. Two cases are then possible. If all the indices appear at least twice in the set, the obtained minimum weight is zero, i.e. the set of gr(x) can generate the zero word with non-zero coefficients. If at least one index appears only once, we may then neglect the multiplicity of the indices and compute the weight of this word using the way described above. Sometimes the BCH bound is below the real minimal weight of a code, as for some quadratic residue codes. When a better bound than the BCH one is known, it can naturally be used in the evaluation of the minimal weight of a word. Tables ILl and IL2 are BCH tables, similar to table I, for other lengths Sequential use of classical bounds Using classical bounds on the minimal weight of each word of a sequence as described in sec. 6., we obtain a lower bound on the free distance of a code, in a way to be now described. Some particular tricks can often be applied if ')I is small, but we now present a general algorithm based on the transfor- n=7 D n n = G ~J n = cf. sec. 6. TABLE ILl n =

10 TABLE IL2 N 0\ 0\ ; n = ; ' ~ S

11 CONVOLUTIONAL CODES AND IRREDUCIBLE IDEÀLS 267 mation 2) of the stack algorithm 7.8). This algorithm is illustrated by the code defined by (6), whose G is rewritten G = [ 7 ~] and whose r is given above (eq. (7». From these matrices we construct two trees (fig. I), which have at each level four new branches for each branch in the preceding level. Each level represents a word and the first level has only the last branches. In the first tree these are indexed, and -, and correspond to the case where the first word is a multiple of gs(x) or of gl(x) or only of their greatest common divisor. The first branch of the other levels corresponds to the case where the infor- _ mation symbols of the corresponding word are zero. Near this first semi-infinite tree, we now construct step by step a similar tree whose branches are indexed by the minimal weight of the corresponding sequences as follows. The three possibilities for the first word correspond to minimal weights that are respectively 6, 6 and 2, as is obtained from sec. 6.. These weights I Fig.. Trees for the computation I 2 r 20 8 of df'

12 268 Ph. PIRET are indicated on the first level of the 2nd tree. The least of these weights is 2 and the corresponding branch must be extended. The interesting submatrix of ris [ ~J ' and the four possibilities for the 2nd word are "weighted" following sec. 6. to obtain as partial weights 2, 6, 8 and 6 and as total weights 24, 8, 20 and 8. The minimal weight of the not-extended branches is now 6 and the corresponding branches are extended. This procedure is continued until a node of minimal weight has an infinite number of successors with the same weight. This weight is a lower bound on the free distance of the code. For the above example, the free distance finally obtained is 6. However, in this particular case, since the first and the last word in any sequence have at least a weight of 2, it is sufficient to show that the total weight of any intermediate subsequence is also at least 2, and this can be achieved by moredirect arguments. 6.. Binary convolutional codes We give here some examples of binary convolutional codes with the obtained lower bound on their free distance. (a) The first codes below can be seen as an extension of the maximal-length sequence block codes n k ')I G d f 2 [, ] ~8 * 2 [, 7] ~ 6 * 4 [,,, ] ~ 64 * [,,,, ] ~76 6 [,,,,, 7] ~84 4 [,,,] ~ 28 * [,,,,2] ~ 4 6 [,,,,2, ] ~ 68 All codes indexed by * meet the Plotkin bound obtained by Layland and McEliece ). (b) The quadratic residue codes are well suited to define good codes with ')=2.

13 CONVOLUTIONAL CODES AND IRREDUCIBLE IDEALS 26 n k G [,] [, ] [,] [, ] d f ~ 2 ~ ~ ~20 A more complete table can be constructed from the one given by Berlekamp ) for QR block codes. (c) General BCH-like convolutional codes are easily constructed and not too difficult to be checked for their d f as explained in secs 6. and 6.2, where a first example was used. We give here some other examples: ~24 n k 'V G 7 7 [; : I~J [:! I: I:J [ J 27 ~ 66 In one of these matrices there appear indexed numbers such as 7a> Sa>Sb. The meaning of these indices refers to the actual generator matrices of the corre- ~24

14 270 Ph. PlRET sponding irreducible codes. For example, 7a denotes the same generator matrix, both times it appears; a and b denote two different generator matrices such that the substitution a -- b leaves no vector of the code invariant except zero. The non-indexed numbers denote arbitrary generator matrices. 7. The construction of asymptotically good convolutional codes The pseudo random encoding used by Justesen 26), is also applicable to the RS-like convolutional codes whose length is (2"'/_ I) (cf. secs 4 and ). We shall only illustrate this by an example. The general discussion about rates, puncturing and optimalization, seems to be only a generalization of Justesen's work based on the above material, and will not be given here. We consider a. primitive RS-like convolutional code on GF(2"') with k = 2 and v ~ 2",-. The jth symbol of each word of this code is then encoded by the jth code in the Wozencraft's ensemble of the pseudo randomly shifted (2m, m), binary codes. We now lower-bound df> the free distance ofthe resulting binary convolutional code. Theorem 7.. When m grows to infinity, the free distance of the binary convolutional code constructed hereabove is lower-bounded by In (20), H-l(X) denotes the inverse of the binary entropy function, univocally specified by 0 ~ H-l(X) ~ t, and nb is 2 m (2"'- ), i.e. the length of a word of the binary convolutional code. Proof Justesen has proved that the linear binary block code obtained by the pseudo random encoding ofan RS block code on GF(2"'), oflength n = 2'"_ I, and rate kin, has a minimal weight that is lower-bounded by nb[h-l(t) - O(m)] (- k/n). As in the proof of theorem 4., we now consider an information sequence of u words that is encoded in Cu + 'ji- ) words. Suppose first that ~ u ~ 'ji-. For all i ~ u-i, the ith and the (u + 'ji- i)th word of the RS-like code are words of an RS block code whose rate is i 2 /(2"' - ). The binary words obtained thus have a weight that is lower-bounded by (20) In the same 'way the ('jiis lower-bounded by u + ) intermediate words have each a weight that nb[h-l(t) - OCm)] [- u 2"/(2"'- )]. The total weight of the sequence is thus lower-bounded by summing the contributions of all these words, to obtain the following expression:

15 CONVOLUTIONAL CODES AND IRREDUCIBLE IDEALS 27 nb[h- (t)- O(m)] {v- + u [- v2 S /(2"'- I)]}. / (2) A similar expression can also be obtained if u ~ v, as in theorem 4.. It is then easily seen that the expression (2) has the form claimed by (20). Q.E.D. If we denote by h(v) the expression (v- ) H- (t), which does not depend on nb' we obtain for the code we just constructed: lim (df/nb) ~ h(v). m-+oo (22) 8. Conclusion In the preceding sections we tried to generalize some classical results on cyclic block codes to obtain constructive families of convolutional codes of sufficient quality. There seem however to remain many open questions in this area, and particularly in the construction of high-rate RS-like codes and of binary BCHlike codes. MBLE Research Laboratory Brussels, May 72 REFERENCES ) R. C. Bose and D. K. Ra y-ch au dh u r y, Information and Control, 68-7, 60. 2) R. C. Bose and D. K. Ra y-ch a udh u ry, Information and Control,27-20, 60. ). S. Reed and G. Solomon, J. Soc. indo appl. Math. 8, 00-04, 60. 4) W. W. Peterson, Error correcting codes, The M.LT. Press, Cambridge, Mass., 6. ) J. L. Massey, Threshold decoding, The M.I.T. Press, Cambridge, Mass., 6. 6) A. D. Wyner and R. B. Ash, IEEE Trans. Inform. Theory IT-, 4-6, 6. 7) K. S. Zigangirov, Problemy Peredachi Informatsii 2, no. 4, -2, 66. 8) G. D. Forney, Concatenated codes, The M.I.T. Press, Cambridge, Mass., 66. ) J. P. Ro binson and A. J. Bernstein, IEEE Trans. Inform. Theory IT-, 06-, 67. 0) S. M. Reddy and J. P. Robinson, Information and Controll2, -70, 68. ) S. M. Reddy, Information and Control, 7-62, 68. _ 2) S. M. Reddy and J. P. Robinson, Information and Control, 42-07, 68. ) E. R. Berlekamp, Algebraic coding theory, McGraw-Hill Book Company, 68. 4) L. J. Massey and M. K. Sain, IEEE Trans. Comp. C-7, 0-7, 68. ) P. M. Ebert and S. Y. Tong, Bell Sys. tech. J. 48, , 6. 6) J. M. Goethals, Codes linéaires définispar des polynornes, Thesis, Université Catholique de Louvain, September 6. 7) D. J. Costello, IEEE Trans. Inform. Theory IT-S, 6-66, 6. 8) F. Jelinek, J.B.M. J. Res. Dev., 67-68, 6. ) J. Layl an d and R. MeElieee, An upper bound on free distance of a tree code, Jet Propulsion Laborat. Calif. Institute Techn. Pasadena, Space Program Summary 7-62, vol., Apr. 70, pp ) G. D. Forney, IEEE Trans. Inform. Theory IT-6, , 70. 2) C. D. Forney, IEEE Trans. Inform. Theory IT-6, 7-7, ) L. J. Massey and D. J. Costello, IEEE Trans. Comrn. Techno!. COM", 806-8, 7. 2) L. R. Bahl and F. Jelinek, IEEE Trans. Inform. Theory IT-7, , 7. 24) J. L. Massey, D. J. Costello and J. Justesen, Polynomials weights and code construction, to be published in IEEE Trans. Inform. Theory. 2) J. L. Massey, Lectures on convolutional codes, given at the Catholic University Leuven, ' January ) J. Justesen, Constructive, asymptotically good, algebraic codes, IEEE Intern. Symposium on Information Theory, Asilomar (Cal.), January 72.