ccsds 131.0-b-2

CCSDS RECOMMENDED STANDARD FOR TM SYNCHRONIZATION AND CHANNEL CODINGWith this method of interleaving, the original kI consecutive information symbols thatentered the encoder appear unchanged at the output of the encoder with 2EI R-S checksymbols appended.4.4.2 DUAL BASIS SYMBOL REPRESENTATION AND ORDERING FORTRANSMISSIONEach 8-bit Reed-Solomon symbol is an element of the finite field GF(256). Since GF(256) isa vector space of dimension 8 over the binary field GF(2), the actual 8-bit representation of asymbol is a function of the particular basis that is chosen.One basis for GF(256) over GF(2) is the set ( 1, α 1 , α 2 , . . ., α 7 ). This means that anyelement of GF(256) has a representation of the formwhere each u i is either a zero or a one.u 7 α 7 + u 6 α 6 + . . . + u 1 α 1 + u 0 α 0Another basis over GF(2) is the set ( 1, β 1 , β 2 , . . ., β 7 ) where β = α 117 . To this basis thereexists a so-called ‘dual basis’ (l 0 , l 1 , . . ., l 7 ). It has the property thatTr(l i β j ) = ⎩⎨ ⎧ 1 if i = j0 otherwisefor each j = 0, 1, . . ., 7. The function Tr(z), called the ‘trace’, is defined by7Tr(z) = ∑ z 2kK=0for each element z of GF(256). Each Reed-Solomon symbol can also be represented aswhere each z i is either a zero or a one.z 0 l 0 + z 1 l 1 + . . . + z 7 l 7CCSDS 131.0-B-2 Page 4-8 August 2011