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Solving Error-Locator Polynomials

The only nonlinear step in the decoding of BCH codes is finding the zeroes of the error-locator
polynomial . If the number of errors, , is not larger than the error correcting ability of the
code, then the degree of is . Polynomials of small degree can be solved using various ad
hoc methods. In particular, polynomials over GF(2m) of degree ≤4 can be factored quickly using
linear methods.

Quadratic polynomials

First we consider an error-locator polynomial of degree 2:



Squaring in GF(2m) is a linear transformation of the vector space GF(2m) over the scalar field
GF(2). Therefore the equation   can be rewritten as



where x is the unknown m -tuple,   and are m×m matrices corresponding to multiplication
by constants and , S is the m×m matrix over GF(2) that represents squaring, and 1 is the
m-tuple (1, 0, . . . , 0). Solutions of this system of equations are zeroes of the error locator polynomial
. The coefficients of the m ×m matrix   involve a matrix multiplication and
can be computed in O(m3) bit operations. The rows of the and matrices can be obtained
by shifts with feedback. The squaring matrix S can be precomputed; its rows are
Once A has been found, approximately bit operations are sufficient to solve the GF(2) system
of linear equations xA = 1. Gaussian elimination uses elementary row operations. Subtracting
two rows of an m ×m matrix over GF(2) can be performed by XORing two m-bit wide registers.
Solving xA = 1 can be done in about row operations. For example, when m = 8, the linear
system can be solved in about 32 clock cycles, compared to 255 evaluations of in the worst
case for trial and error.
Quadratic polynomials can be solved even faster using the change of variables

The transformed equation is of the form u2 + u + c = 0 or U(S + I) = c. It can be solved using
a precomputed pseudo -inverse R of S + I . If U1 = Rc is one zero, then U2 = U1 + 1 is the other
zero, since the sum of the two zeroes is 1. Thus the zeroes of the original polynomial are

Only half the values of c yield solutions, so we must check that c belongs to the range of S + I .

Another way to solve the simplified quadratic polynomial is to store the zeroes of the simplified
quadratic polynomials u2 + u + c in a table of 2m entries indexed by the constant term c. This
approach is feasible if 2m is not too large. The polynomials u2 + u + c have zeroes in GF(2m) for
exactly half of the elements c in GF(2m). For those values of c for which the polynomial does not
have zeroes, a dummy entry such as 0 can be stored.

Once the error locators X1 and X2 have been calculated, log tables may be needed to convert
to symbol error locations and .

Cubic and quartic polynomials

Cubic polynomials, which arise in triple error correcting codes, can also be solved using lookup
tables of size 2m. A change of variables of the form x = au + b can transform the error-locator
polynomial

to one of two simplified forms:

or

Given a table indexed by c that stores two zeroes U1, U2 of u3 + u + c, the third solution is
For the other form of reduced polynomial, u3 + c, one zero U1 could be stored
in a table; the other zeroes are and , where
γ is a cube root of 1. (When m is
odd, every element of GF(2m) has a single cube root, hence u3 +c has one zero of multiplicity 3.)
Cube roots can also be easily computed using logarithms.

Cubic and quartic polynomials over GF(2m) can also be solved using linear methods. The
error-locator polynomial of degree 4,

can be reduced by a change of variables of the form x = (au + b)-1 to a polynomial

The zeroes of this polynomial are the solutions of the linear system

where S is the squaring matrix.

Polynomials over GF(2m) can be factored efficiently using Elwyn Berlekamp’s factoring algorithm
(Bell System Technical Journal, 46:1853-1859, 1967). The running time for polynomials of
degree n is for a small exponent c , but the constant factor is too large for many applications.

Chien search


Using Horner’s method, any polynomial of degree can be evaluated at a single point using
multiplications and additions. Viktor Pan proved in 1962 that multiplications are needed in general.
However, evaluating a polynomial at n points can sometimes be done using much less than
operations. For example, the discrete Fourier transform, which consists of evaluating a polynomial
(the input vector) at all n-th roots of 1, can be computed using O(n log n) multiplications
and additions.

The Chien search is a clever method for solving the error-locator polynomial by brute force—
we evaluate for every i=1,2,...n. The Chien search evaluates for successive values
of i using constant multiplications instead of general multiplications. (This is surprising,
because the coefficients of the error-locator polynomial are not known in advance.)
Furthermore, each evaluation of takes place in one clock cycle instead of clock cycles—
without resorting to pipelining techniques.

The key idea of the Chien search is to maintain state variables, Qj for j = 1, . . . , , where
at every time i. Each state variable is easily updated by multiplication by a constant:
. The sum of these state variables is , so an addition tree is sufficient to tell
when

A Chien search circuit is shown in the following figure. In this circuit, and in the modified
Chien search circuits shown later, the memory elements are initialized with the coefficients of the
error-locator polynomial. When , we set for

The output signal ERRLOC is true when . Since the zeroes of are the reciprocals
of the error-location numbers, ERRLOC is true for those values of i such that   is
an error-location number. Thus as i runs from 1 to n, error locations are detected from the most
significant symbol down to the least significant symbol.

In the binary case—channel alphabet GF(2)—the ERRLOC signal can be used directly to
correct the received bit In the general case, the error pattern Yl must be computed from the
syndrome equations or by Forney’s algorithm, which is given later in this handout.

The Chien search is often fast enough because the number of clock cycles it requires is very
close to the number of information symbols. Thus error correction can be performed in the time
it takes to compute the syndrome of the next received n-tuple.

A straightforward way to speed up the Chien search is to run two searches in parallel. The
following circuit is more efficient than two separate copies of the Chien search engine because the
memory storage elements are shared.

In the above circuit, ERRLOC0 is true when an even error location is detected, while ERRLOC1
indicates odd error locations. (The additional combinational logic may require a longer clock
cycle.) The scaler usually requires more gates than the scaler for small values of j. Thus
the complexity of a double-speed Chien search circuit can be further reduced by factoring the
scalers as

The combinational logic delays of this circuit may be slightly greater than those of the first
double speed Chien search circuit because the optimized logic for multiplying by may be faster
than the sum of the delays through two scalers.

In practice, the Chien search is often run backwards, using scalers for The
reverse search locates errors starting with error location 0. When the received data is already
stored in random access memory, there is no advantage in correcting the symbols starting with
the most significant symbol

The error magnitudes Yl , which are given by the Forney algorithm,

can be computed by a circuit similar to the Chien search circuit that operates in parallel with the
Chien search. At each time i, the values of and are available, so Yl can be computed
with one division in GF(2m) and one multiplication by . Since b is a constant, successive
values of can be computed by multiplication by the scaler . The multiplication
by   is not needed when b = 1, which is a minor advantage to using narrow-sense BCH
codes.

Although the total number of operations performed is greater than simply evaluating the polynomials,
the Chien-style circuit uses low-cost scalers and has very simple control logic. Another
advantage of this approach is that each error magnitude can be computed quickly after the corresponding
error location is found. For the narrow-sense BCH code, one division operation is needed
for each corrected error.

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