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AN2407

Reed Solomon Encoder/Decoder on the StarCore™ SC140/SC1400 Cores, With Extended Examples

Freescale Semiconductor 

Results

• Roots Search. An evaluation of Λ(x) on all field points. Each of the 256 results is subsequently

screened for zeroes. M in this kernel is equal to 256 and D is equal to T+1. If the split-summation

method is applied similarly to the syndromes-calculation kernel, a second power matrix, namely the

transpose of the exp_table_for_syndrome matrix, is required. To conserve memory, we simply invert

the order of the loops in code Example 1 and apply the multi-sample method instead of split-

summation. This is conceptually equivalent to interchanging the roles of M and D. Furthermore, unless

T + 1 is a multiple of four, the split-summation method wastes many cycles on useless calculations.

Therefore, an additional benefit gained by choosing the multi-sample method is that the cycle count

becomes directly proportional to T. The price of using the multi-sample method is that there are not

enough registers to complete the inner loop in five cycles. In this case, the inner loop requires six

cycles instead of five. The lower bound on the cycle count is 384T + 384 cycles, not counting cycles

required for zero-screening. The zero-screening procedure is performed by reading the results byte-

wise and testing for zero. This is performed two cycles per byte and thus requires 512 additional

cycles. Summarizing these factors, the lower bound on the cycle count is 384T + 896 cycles, not

including overhead.

• Forney algorithm. A series of three concatenated polynomial evaluations. However, the entries into

the power table are the error locations, which are randomly distributed. Thus, the power table is not

read continuously from the power table. This adds two cycles to the inner loop. The cycle count, not

including overhead, is at least:

7(nerrors + T)

T--

4

+ 7nerrors

-n---e--r--r--o--r---s

4

+ 6T

5 Results

This section presents the cycle count simulation results for the various decoder routines. All results in this section

are for the case of T = 8.

Syndromes Calculation

6000

Berlekamp-Massey Algorithm

4000

4000

2000

3000

2000

1000

0

0

4000

3000

2000

1000

0

0

2

4

6

Chien Search

2

4

6

0

8

0

2

4

6

8

Forney Algorithm

600

400

200

0

8

0

2

4

6

8

Figure 7. Average Cycle Count of Decoder Routines, as a Function of nerrors

Reed Solomon Encoder/Decoder on the StarCore™ SC140/SC1400 Cores, With Extended Examples, Rev. 1

Freescale Semiconductor

17

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