How to create QRcode page3

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2.3 Calculating error correcting code words

Reed-solomon error correcting is used in QRcode.

First, result data in previous section are delimited to RS block data by rule of table5.
In example data, RS block number in 1-H is 1, we need not delimit.

Next select g(x) from table3.
In example data,count of error correcting code words is 17, we select below g(x).

g(x)=x¹⁷ +α⁴³x¹⁶ +α¹³⁹x¹⁵ +α²⁰⁶x¹⁴ +α⁷⁸x¹³ +α⁴³x¹²
    +α²³⁹x¹¹ +α¹²³x¹⁰ +α²⁰⁶x⁹ +α²¹⁴x⁸ +α¹⁴⁷x⁷ +α²⁴x⁶
    +α⁹⁹x⁵ +α¹⁵⁰x⁴ +α³⁹x³ +α²⁴³x² +α¹⁶³x +α¹³⁶

Above α is a primitive element on GF(2⁸).

Features of GF(2⁸) are ...
 1.four arithmetic operations are supported
 2.α²⁵⁵=1
 3.We can convert exponential in α to integer (or vice versa) using table4.

Now polynomial f(x) which coefficients are data code words is divided by g(x).


f(x)=32x²⁵ +65x²⁴ +205x²³ +69x²² +41x²¹ +220x²⁰ +46x¹⁹ +128x¹⁸ +236x¹⁷ <---(1)

divide by g(x)

Coefficient of leading term in f(x) is 32.
For 32 is α⁵ from table4, we use
 g(x)*(α⁵)*x⁸

=α⁵*x²⁵ +α⁵*α⁴³*x²⁴ +α⁵*α¹³⁹*x²³ +α⁵*α²⁰⁶*x²² +α⁵*α⁷⁸*x²¹.....

=α⁵*x²⁵ +α⁴⁸*x²⁴ +α¹⁴⁴*x²³ +α²¹¹*x²² +α⁸³*^x21.....

=32x²⁵ +70x²⁴ +168x²³ +178x²² +187x²¹..... <---(2)

calculate exclusive logical sum (1) and (2)

f(x)'=7x²⁴ +101x²³ +247x²²+146x²¹.....

We repeat same logic until this devide calculation is over. Next, for 7 is α¹⁹⁸,we use g(x)*α¹⁹⁸*x⁷
If exponent of α is over 255,then we decrease it using α²⁵⁵=1

Finally we can get below remainder R(x).

R(x)=42x¹⁶ +159x¹⁵ +74x¹⁴ +221x¹³ +244x¹² +169x¹¹+239x¹⁰
　　　 +150x⁹ +138x⁸ +70x⁷ +237x⁶ +85x⁵ +224x⁴ +96x³ +74x² +219x +61

(see table6)

So we can get

32 65 205 69 41 220 46 128 236 42 159 74 221 244 169 239 150 138 70 237 85 224 96 74 219 61

next page : Data allocation.

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