Table-Driven Decoding of Binary One-Half Rate Nonsystematic Convolutional Codes

Table look-up based decoding schemes for convolutionally encoded data have been designed for both block coding [5], and convolutional coding [6, 21. However, in the latter case most of the work was done for systematic codes. Nonsystematic codes offer better error correcting capability than systematic codes if more than one constraint length of received blocks are considered [7J. Our approach to table look-up based decoding of nonsystematic convolutional codes was introduced in [l, 31. A l/2-rate convolutional encoder is characterized by two generator sequences g(j) = (gf),g;j), ..., gp)), j = 1,2, where U is the constraint length of the code, i.e., the number of memory elements in a minimal realization of the convolutional code [4]. The output conatmint length i s defined as = Z(u + 1) and is equal to the number of encoded bits affected by a single input information bit [TI. An input information sequence U is encoded into two encoded output sequences v(j), j = 1,2, using v = uG, where v = ( u p ) , up), up), up), ...) is the composite encoded sequence, also called a codeword, obtained by multiplexing the two encoded sequences, and G is the semi-infinite code generator matrix [?I. The encoded bits in l/a-rate coding are generated at twice the input information rate. However, it is possible to find an encoding operation that relates blocks of input information bits to equal length blocks of encoded bits [I, 31. Proposition: For I/%-rate convolutional coding with constraint length v, there exists a correspondence between equal length blocks of input information bits and the encoded bits. The length of these corresponding blocks is 2v bits. (For proof, sec [l, 31.) We can formalises this relationship as follows. Let [ul;,i+zu-l=(ui, 2;i.j.,, .q+a.-x{ be a Zv-bit block of the input information sequence, . ,a,+g-l= uzi, uai+t, ... , uai+aV-l) be the 2u-bit block of the corresponding encoded sequence. Given the generator sequences &), j = 1,2, for a l/Z-rate convolutional code with constraint length U, we define the reduced encoding matrix as