Algebraic Soft-Decision Techniques for Linear Block Codes

Thls paper presents a way of incorporating sofl-decision information as an integral par! ofthe pro­ cess of decoding linear block codes which nave an algebraic structure. The quantised demodulator output leveis of a soft-decision communication system are represented by the elements of a Galois field. As a consequence it is then possible to define a 80ft syndrome, a 80ft standard array and a 80ft trellis for decoding linear block codes. This approach stands in opposition lo deconding me­ thods which combine hard-decisions with the digil reliability measures ln an ad-hoc fashion. A 80ft version of an error-trapping decoder for cyclic codes is given as an application of the theory deve­ loped. The basic BCH (Bose-Chaudhuri-Hocquenghem) algebraic algorithm needs to be rnodified in order to benefit trom the procedures here described.


Introduction
This paper investigates a way of incorporating soft-decision information as an integral part of the process of decoding linear block codes [1] which have an algebraic structure.This approach stands in contrast to decoding algorithms which employ conventional hard-decision and, as separate items, the digit reliability measures are used in an ad-hoc fashion [2] and [3].Some practical savings can therefore be obtained by eliminating the need to combine hard-decisions with digit confidence measures in order to achieve soft-decision decoding when such algorithms are employed.Due to their practical importance only binary codes will be considered in the sequei.However, the concepts introduced can be easily extended to p-ary alphabets, p~2.
Soft-decision decoding of (n, k, d) linear binary block codes is studied when the quantised demodulator output leveis are represented by elements of a Galois field GF(2 m ) [4].Thecode digits are elements of the ground field GF(2) which V.C. da Rocha Jr. is a Professor at Departamento de Eletrônlca e Sistemas, Universidade Federal de ~emarnbuco, SO. 741 Recife PE, Brasil.'.G.Farrel is a Professor at Department of Electrícal Engineering, University of Manchester, Manchestei i1113 9PL.. England.levlsta da SocIedade Brasileira de Telecomunlcaç15es 'olume S, N ~ 1, junho 1990.at the receiver are interpreted as ali-zero and all-one m~tuples, in the absence of noise.This representation actual/y maps the original codewords into a subset of codewords of a multilevel (n, k, d) code over GF (2 m ) which constitutes a subspace.As wil1 be shown below, it is then possible to recalculate parity-checks using the quantised demodulator output leveis directly.The concept of a soft-syndrome then emerges naturally.Also, the decomposition of the softly quantised space of n-tuples into cosets is done with a soft standard array where, contrary to the hard-decision case, some coset leaders are allowed to have the same soft syndrome and still be unmistakably correctable.A soft trellis is then introduced which is actually the trellis [5] for the multilevel code over GF(2 m ), and although it could be used in principie to decode the binary code, it serves perhaps a better purpose in showing the relationship between these two codes.The next obvious step is to investigate the behaviour of soft-decision decoding algorithms which were previously used for the hard-decision decoding of block codes and, as will be shown, are able to process multilevei symbols with only a minor modification.A soft error trapping (ET) decoder [6] is described as an application of the theory developed.A slight modification in the basic BCH algebraic decoding algorithm [7] is introduced which allows, in some special situations, an efficient handling of error patterns that cannot be corrected by a hard-decision decoder.

Channel Output Quantisation
ln practice, very often the analogue voltage at the demodulator output of a digital communication system is quantised.Also, it is common to choose the number q of quantisation leveIs to be a power of 2, i.e., q = 2 m in order to simplify further digital processing [7].This paper is not concerned with problems like optimal quantisation leveI spacing and choice of metric.ln the sequei equally spaced quantiser thresholds and integer metric values from Oto 2 m -1 will be assumed.This choice leads to some impairment in performance which is minor however when eight or more quantisation leveis are used 17].
The q = 2 m integer metric values have been represented traditionally by binary m-tuples [8].Other representations are possible of course.
Though known not to be an optimal procedure it is acceptable to label the 2 m leveis with the integers O to 2 m -1 because the impairment in performance is only minor [7].Since there is a one to one correspondence between binary m-tuples and the elements of GF(2 m ) it has been suggested by the first author [9] to represent the demodulator output leveis by the assocíated Galois tield elements.Such a representation is not obvious since the idea of order is nonexistent in a Galois field.That is difficult to reconcile with the definition of soft-decision distance [8] which states that the soft-decision distance between

60
. J two leveis is given by the absolute value of their difference.This apparent impasse is removed by observing that for fields of characteristic 2, i.e., GF(2 m ), the integer value corresponding to the modulo 2 difference between any two field elements coincides with their soft-decísion distance when one of them is either the ali-zero ar all-one m-tuple.'ln general, the integer value of the difference between any two field elements may not coincide with their soft-decision distance.

Example 1
Consider q=4 quantísation leveis.The elements of the Galois field GF(4) are O, 3 For any two leveis, it is easily verified that the integer value corresponding to their modulo 2 difference (which is the same as modulo 2 addition) coincides with their soft-decision distance only when one of the leveis is either 00 or 11.
ln the process of decoding binary codes the components of the softly-quantised received n-tuple are always compared against top confidence leveis, i.e., ali-zero ar all-one m-tuples, which are the components of the valid codewords.Therefore, in this case, the levei assignment proposed above is perfectly satisfactory.The above considerations are summarized in the fol/owing /emma.

Lemma 1
For fields of characteristic 2, i.e., GF(2 m ), the integer of the modulo 2 difference between a pair of elements in the field concides with the value of their soft-decision distance when one of the elements considered is either the ali-zero or the all-one m-tuple.
The soft-decision levei representation suggested above allows the integration of soft-decision information as part of the shift and add operations of an error-trapping (ET) decoder, as shown later in this paper.

80ft 5yndromes for Decoding Linear Codes
1t is well known in coding theory that the decoding problem can be approached in principIe either by using the maximum Iikelihood criterion, which is codeword oriented, or the syndrome, which is error-pattem oriented.80th approaches have advantages and disadvantages and the eventual choice for one of them depends on the characteristics of the code to be used.ln soft•decision decoding, most of the practical approaches to the problem have shown two common points.First, the decoder operates on hard-decision estimates of the demodulador output forming a hard-decision syndrome and, as a separate tool, uses the reliability measures associated with the hard decisions to assist on the cc;>rrection of errors.Very often the reliability measures are combined with the hard-decision estímates in an ad-hoc manner [2] and [3].Second, the more sophisticated soft-decision decoding algorithms [5] and [10] are directed to the esti mation of bits 01' codewords inste<::c: of error-patterns.
The concept of a soft syndrome becomes plausible from the mcment one can operate with the softly quantised symbols of the received n-tuple in order to recalculate parity-checks.The idea is to effectively integrate the demodulator' output leveIs into the decoding processo The following example serves to illustrate this poinl.
Example 2 Consider the (3, 2, 2) ~)jnary single-parity-check code and q=4 quantisation leveis.The code parity-check is given by c =k1 + k2.The demodulator output leveis are assumed represented as in Example 1. Suppose the message to be transmittted is 10.Its associated codeword is 101.ln the absence of noise it is received at the demodulator output as (a 2 ,0, ( 2 ).Now suppose that due to noise the received word is (a,O,a 2 ).The recalculated parity-check is a+O = a.

50ft Standard Array
The standard array [1] is a well known concept in coding theory, commonly presented in conjunction with hard-decision decoding procedures.As a conceptual tool a soft standard array can be useful in throwing some Iight on the cade structure and decoding processo The soft standard array contains 62 ~ -----~ n-tuples which satisfy the code paríty-check equations but are not in its top row and therefore do not belong to the (n,k,d) binary cade.Such n-tuples are called soft cadewords and in general may be seen as belonging to a (n,k,d) code with a higher order alphabet q = 2 m which contains as a proper subset the soft-decision mapped (n,k,d) binary code.As a consequence, contrary to the hard-decision case, many correctable error patterns have the same syndrome.precisel Y ( the soft standard array has 2 k columns and 2 mnk rows such that sets of 2 m-1)k rows have the same soft syndrome.

Example 3
The soft standard array for the (3,2,2) code of Example 2 is The correctable soft error patterns are the coset leaders of the top eleven rows.

Soft-Trellis Decoding of Block Codes
A trellis constructed along the same Jines as indicated by Wolf [5) considering ali the paths which satisfy the cade parity-check equations with symbols trom GF(2 m ) is called a soft trellis.The soft trellis for the (3,2,2) binary cade of _AClllllJltI é.IS glVen in Fig. 1 together with the hard-trellis for comparison.The decodlng operations are identical to those of a Viterbi decoder [7].ln a soft-decision receiver, using 2 m quantisation leveis, in the absence of noise the received codeword leveis would be of top confidence, i.e., either ali-zero or all-one m-tuples.Therefore the original binary codewords of the (n,k,d) cyclic code appear mapped as a proper subset of the 50ft cyclic code.
However, the minimum distance between valid codewords is now d s =(q-1)d rather than d [7].The concept of a 50ft cyclic cade is very interesting because it allows the tools developed for linear codes to be used in a soft-decision cantext For example, the decomposition of the soft1y quantised space of n-tuples into cosets can be done with a soft standard array, syndromes can be calculated, etc.Some theoretical tools are next introduced in the form of lemmas and theorems.They will be useful in applícations presented later.

Proof
By definition ali codewords of the soft cyclic code satisfy the binary code parity-eheck equations and therefore are multiples of the code generator polynomial g(x).

Lemma 3
The m-1 elements, ai, O ~i~ m-2, of GF(2 m ) are linearly independent and by linear combinations only generate elements of soft weight less than (21TL.1 )/2.

Proof
The first par! is a well known result since, actually, the m elements, ai, O ~i ~ m-1, of GF(2 m ) are Iinearly independent [4].For the second par!, considering that the field characteristic is 2, the linear combinations of the ai, O ~i~ m-1, can be thought of as modulo 2 sums S of m-1 bit binary numbers.Thus, the AI this point it is convenient to describe a decomposition of the 50ft cyclic code which will be called a soft cyclic code array (SCA).The SCA is a rectangular array with rows and columns.Its top row is filled with the top confidence 50ft codewords, beginning with the ali-zero n-tuple, i.e., as the lea der of the first row.The leader of the second row is chosen to be a previously unused soft codeword of 10west soft weight and the remaining row places are each one filled by adding its leader to the top confidence codeword situated in the same column, i.e., ímmedíately above.The third and successive rows are formed in a similar manner, beginning with a previously unused soft codeword of least soft weight.As a result, a total of 2(m-1)k rows are thus formed.

Theorem 1
The SCA leaders have soft weight W ~ts and therefore represent correctable soft error patterns.

Proof
Suppose a leader Lhas soft weight W ~ts + 1.Since the top row of the SCA contains words of 50ft weight d s ~2ts + 1, this implies the existence of at least one 50ft codeword of soft weight less than or equal to t s in the row where L is the leeder.Therefore e contradiction to the SCA construction mies.

Tt'leorem 2
Any soft codeword can be written as a sum, over GF(2 m ), of a top confidence 50ft codeword and a 50ft codeword of 50ft weight at most ts.

Proaf
This is an immediate consequence of the SCA construction combined with the r6sult of Theorem 1.

Proaf
By definition the soft cyclic code has k information positions which are ocwpied by elements of GF(2 m ).Thus, a total of 2 mk soft codewords resu!t.
The SCA has 2 k columns and therefore 2 mk /2 k = 2(m-1)k rows, Le., 2(m-1)k leaders.It is now an easy matter to deduce that the first column of the SCA forms a subspace, Le., a linear code, of dimension k whose information positions are occupied by the m symbols (O, 1, a, a 2, •••, a m -2) and by Lemma 3 the theorem follows.

heorem 4
A binary hard-decision version of a received soft codeword gives the right transmitted codeword when the soft error patterns is a correctable one.

Proaf
Conceptually, the decoding of a soft codeword 1) can be performed as follows.
First locate the SCA row where 1) js situated and then subtract the row leader r from 1).The result is a top confidence soft codeword c.Since d s :o;;;2t s + 1 and by Theorem 2 and Lemma 3, the row leader r, whose individual components have soft weight less than q/2 and of total soft weight at most ts, is unique and is thus assumed to be the soft error pattem.Therefore, the nearest binary codeword is a hard-decision version of 1).

Soft Error-Trapping Decoder
!n this section a soft-decision versionof an error-trapping (ET) decoder for cyclic codes is described.The ET decoding technique [1] is very simple and is usually applied for the hard'"CIecision decoding of cyclic codes.It is most efficient for rate R< 1/t codes, where t is the rnaximum number of errors to be corrected [1].Modified ET decoders have been proposed which partially avoid this limitation for codes not satisfying R< 1/t [11].Nonbinary ET decoders have been previously proposed [12], however to deal with hard-decision decoding of multilevel cyclic codes, Assume the channel output quantisation is performed as described above.The softly-quantised received n-tuple is fed into a GF(2 m ) division circuit wired for division by the code generator polynomial g(x).As a result the soft syndrome is formed.The syndrome soft-weight (SW) is compared to a threshold t s • where l (2m -1) d -1 j t s os

b) SW >t s
The syndrome ib shifted inside the divísion circuit, with feedback on.After each shift the updated syndrome weight SW is compared wíth ts' If SW :s;;t s after the i th shift, 1 "'i i :s;; n-1, then a soft error has possibly been trapped.The decoder subtracts the ;:;hifted syndrome fram the i th cyclic shift of the received r,..tuple.If the resu1t is a top confidence codeword, i.e., a codeword whose elements are represented by ali-zero or all-one m-tuples, then stop.The error has been trapped and the received word has been corrected.Otherwise, continue the process of shifting the syndrome and comparing SW to t s .If after n shifts either SW has never been equal to or less than t s or, if that happenned and the attempted correctiondid not produce a top confidence codeword, then an error has been detected which cannot be trapped.
The theoretical basis fel the above procedure is now described.

Theorem 5
Let a correctable error pattern with SW ~ts affect the transmitted codeword.
Then one of the folJowing situations will occur.
a) SW =O a 1) No errors.a2) An error which coincides with a soft codeword whose individual components have soft-weight less than 0/2.

b) 0< SW < t s
If each syndrome component has soft weight less than 0/2 then a soft error pattern is assumed, where each nonzero position has soft weight less than q/2.Otherwise nothing can be saído c) SW =is Ali the errors occur in the parity-check section of the received n-tuple.

d) SW < t s
The errors are not ali in the parity-check section of the received n-tuple.

Proa'
Case a is obvious.Case b can be established with the help of Lemma 3. Case c follows by reasoning that if the assertion made is false, i.e., the errors are not confined to the word parity-eheck positions, then by subtracting the syndrome from the received n-tuple would cause a total of at most 2t s soft-weight changes in the transmitted codeword and the result would be a valid top confidence codeword.Sut that is impossíble because d s ;:;;,2t s + 1 is the minimum number of soft-weight changes needed to produce a valid top confidence codeword.Case d follows trivially from case G.

Example 4
Consider the (7,4,3) binary cyclic Hamming code with generator polynomial g(x) = x~ + x + 1.The demodulator output is quantised into 8 amplitude leveis, represented below by the powers of a primitive element a pf GF(B), where the element a was chosen to be a root of the primitive polynomíal x~ + x + 1.
The code soft-decision distance is given by d s = (8-1 ).3 = 21, therefore t s = 10.Suppose the ali-zero codeword is transmitted and the received word in polynomial form is a 4 x 6 + a 2 x 5 , i.e., and error of soft weight 6+4 = 10.The received polynomial is divided by g(x) and the remainder, which is the soft syndrome, is found to be ax 2 +a 2 x + a.The evolution of the decoding process is illustrated below, where (c, b, a) represents the coefficients of the polynomial ax 2 +bx+c.

SHIFT
• ..... (a 2 , a 4 , O) The soft syndrome has weight 8 which is less than t s = 10 but since its components are not ali of soft weight less than q/2=4 the decoder proceeds.
After the first shift, SW = 6 < ts.The received n-tuple is shifted once.
Correction is attempted by subtracting the syndrome from the shifted version of the received n-tuple.Since the result is not a top confidence codeword the decoding operation continues.ln the second shift SW equals t s and correctíon is attempted again.This time it succeeds, i.e., the result of subtracting the Revista da Sociedade Brasileira de Telecomunicações Volume 5, Ne 1, junho 1990.
syndrome trom the received n-tuple cyclically shifted twice gave as a result a top confidence codeword.The decoding operation is halted.The decoder registers are c1eared and wait for the next received word.Notice that such an error pattem wnuld not be corrected by a hard-decision decoder.For this (7,4,3) code it is clear that some double-error patterns cannot be trapped since R > 1/t, i.e., 4/7 > 1/2.

BCH Decoding
One snag with the algebraic decoding of BCH codes has been its error-correction bounding by the designed distance rather than by the minimum distance.The limitation in error-eorrection capability of the basic algebraic algorithm possibly results ber.ause it makes no essential use of• the fact that the binary (n,k,õ) BCH code is a subfield subcode of a Reed-Solomon code with the same õ hawever over a higher order alphabet.The decoding algorithm being exartly ldentical for both codes.Some of the ideas developed in this paper may be of relevance here.1f the frequency domain decoding approach is used [7] then it follows that for a given õ there are &-1 known consecutive coefficients of the finite field Fourier transform of the errar polynomial and therefore a recursion relation involving an elTlJr locator polynomial of degree at most õ-1 could in principIe be established.Both the Berlekamp-Massey and Euclídean algorithms [7] only provide error 10cator polynomials of degree at most l(&-1)/2J and thus are not applicablp.in the soft decision case.By using the received digit reliability measures a most Iíkely error locator polynomial could be formed.The rerursion relation is tested for consistency.If it works then the correction of the errors praceeds in the normal way.If it fails then it is possible that olle or more 'errors of top confidence have hit the received codeword.ln this case some algebra has to be used in arder to solve the problem.Unless õ is a small number the complexity of such an approach becomes prohibitive.

Comments
The procedures described above are not restricted to codes of lenght n = 2 m 1 with q = 2 m quantisation leveis.As m grows it is both impractical and unnecessary to keep the 2 m qU::.inti s3tion leveis.Either a smaller field contained in GF(2 m ) is employed ar, if it is not contained in GF(2 m ), each of its elements can be made to represent a c1uster of elements of GF(2 m ) which are neighbclrs in the soft•jecision sense.The code length in general will be a divisor of 2 m -1.It f'hould be remarked that the ET algorithm described is sul:roptimum bacaus'" of the channel output quantisation, the use of soft-decision distance instead of Euclidean distance, and the fact that ET is not always equivalent to minimum distance decoding.Using a slightly modified version of the ET algorithm presented above, which makes use of covering polynomials [11], it is possible to correct most of the error patterns containing two hard-decision errors by means of binary Hamming codes.

Figure 1 .
Figure 1.Soft trellis and hard trellis carresponding to the cade in Example 2.

2
and lx] means the inteÇjer pari of x.One of the following situations will occur.a)SW ost sThe decoder delivers to the ::ink a hard'"CIecision version of the received n-tuple if each syndrome component has soft weight less than ql2.Otherwise proceed as; in b).