Berlekamp–Massey algorithm: Difference between revisions

Revision as of 18:01, 23 July 2023

Revision as of 01:58, 17 October 2023

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The algorithm also needs to increase ''L'' (number of errors) as needed. If ''L'' equals the actual number of errors, then during the iteration process, the discrepancies will become zero before ''n'' becomes greater than or equal to 2''L''. Otherwise ''L'' is updated and algorithm will update ''B''(''x''), ''b'', increase ''L'', and reset ''m'' = 1. The formula ''L'' = (''n'' + 1 − ''L'') limits ''L'' to the number of available syndromes used to calculate discrepancies, and also handles the case where ''L'' increases by more than 1.

==Code samplePseudocode ==

The algorithm from {{Harvtxt|Massey|1969|p=124}} for an arbitrary field:

<!-- Notes: notation changes from Massey:

Massey Here

Line 103 ⟶ 102:

T(D) T(x) polynomial

-->

<div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr">

<syntaxhighlight lang="c">

<span></span><span class="n">polynomial</span><span class="p">(</span><span class="n">field</span><span class="w"> </span><span class="n">K</span><span class="p">)</span><span class="w"> </span><span class="n">s</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="cm">/* coeffs are s_j; output sequence as N-1 degree polynomial) */</span>

polynomial(field K) s(x) = ... /* coeffs are s_j; output sequence as N-1 degree polynomial) */

<span class="cm">/* connection polynomial */</span>

<span class="n">polynomial</span><span class="p">(</span><span class="n">field</span><span class="w"> </span><span class="n">K</span><span class="p">)</span><span class="w"> </span><span class="n">C</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span><span class="cm">/* coeffs are c_j */</span>

polynomial(field K) C(x) = 1; /* coeffs are c_j */

<span class="n">polynomial</span><span class="p">(</span><span class="n">field</span><span class="w"> </span><span class="n">K</span><span class="p">)</span><span class="w"> </span><span class="n">B</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>

polynomial(field K) B(x) = 1;

<span class="kt">int</span><span class="w"> </span><span class="n">L</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span>

int L = 0;

<span class="kt">int</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>

int m = 1;

<span class="n">field</span><span class="w"> </span><span class="n">K</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>

field K b = 1;

<span class="kt">int</span><span class="w"> </span><span class="n">n</span><span class="p">;</span>

int n;

<span class="cm">/* steps 2. and 6. */</span>

<span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">&lt;</span><span class="w"> </span><span class="n">N</span><span class="p">;</span><span class="w"> </span><span class="n">n</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>

for (n = 0; n < N; n++) {

<span class="w"> </span><span class="cm">/* step 2. calculate discrepancy */</span>

<span class="w"> </span><span class="n">field</span><span class="w"> </span><span class="n">K</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">s_n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><math>\sum_{i=1}^L c_i \cdot s_{n-i}</math><span class="p">;</span>

field K d = s_n + \Sigma_{i=1}^L c_i * s_{n-i};

<span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">d</span><span class="w"> </span><span class="o">==</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>

if (d == 0) {

<span class="w"> </span><span class="cm">/* step 3. discrepancy is zero; annihilation continues */</span>

<span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>

m = m + 1;

<span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="mi">2</span><span class="w"> </span><span class="o">*</span><span class="w"> </span><span class="n">L</span><span class="w"> </span><span class="o">&lt;=</span><span class="w"> </span><span class="n">n</span><span class="p">)</span><span class="w"> </span><span class="p">{</span>

} else if (2 * L <= n) {

<span class="w"> </span><span class="cm">/* step 5. */</span>

<span class="w"> </span><span class="cm">/* temporary copy of C(x) */</span>

<span class="w"> </span><span class="n">polynomial</span><span class="p">(</span><span class="n">field</span><span class="w"> </span><span class="n">K</span><span class="p">)</span><span class="w"> </span><span class="n">T</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">C</span><span class="p">(</span><span class="n">x</span><span class="p">);</span>

polynomial(field K) T(x) = C(x);

<span class="w"> </span><span class="n">C</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">C</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><math>b^{-1} x^m</math><span class="w"> </span><span class="n">B</span><span class="p">(</span><span class="n">x</span><span class="p">);</span>

C(x) = C(x) - d b^{-1} x^m B(x);

<span class="w"> </span><span class="n">L</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">L</span><span class="p">;</span>

L = n + 1 - L;

<span class="w"> </span><span class="n">B</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">T</span><span class="p">(</span><span class="n">x</span><span class="p">);</span>

B(x) = T(x);

<span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">d</span><span class="p">;</span>

b = d;

<span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>

m = 1;

<span class="w"> </span><span class="p">}</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="p">{</span>

} else {

<span class="w"> </span><span class="cm">/* step 4. */</span>

<span class="w"> </span><span class="n">C</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">C</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="w"> </span><span class="o">-</span><span class="w"> </span><span class="n">d</span><span class="w"> </span><math>b^{-1} x^m</math><span class="w"> </span><span class="n">B</span><span class="p">(</span><span class="n">x</span><span class="p">);</span>

C(x) = C(x) - d b^{-1} x^m B(x);

<span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">m</span><span class="w"> </span><span class="o">+</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span>

m = m + 1;

<span class="w"> </span><span class="p">}</span>

}

<span class="p">}</span>

}

<span class="k">return</span><span class="w"> </span><span class="n">L</span><span class="p">;</span>

return L;

</div>

</syntaxhighlight>

In the case of binary GF(2) BCH code, the discrepancy d will be zero on all odd steps, so a check can be added to avoid calculating it.