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In books on Fields, an extension field polynomial representation uses the notation where right most bit is considered as $a_0$ & left most is $a_{n-1}$

For e.g. in $F_{2^4}$

$11 = 1,0,1,1 = x^3 + x + 1$

i.e. it's read left to right as $a_{n-1}x^{n-1} + ... + a_0 x^0$

While in coding theory books, in the chapter on cyclic codes, a cyclic codeword $a_0, .., a_{n-1}$ is represented as the polynomial $a_0x^0 +...+ a_{n-1}x^{n-1}$

i.e. a codeword $1,0,1,1$ would be represented as the polynomial $1 + x^2 + x^3$.

I understand this may be purely notational but was wondering if there is a reason.

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    $\begingroup$ Usually, polynomials $f(x)$ are ordered by degree of the monomials, so that the degree of the polynomial is first: $x^2+x+1$. For cyclic codes, the vector of coefficients is written in the opposite direction, see above. $\endgroup$ Commented Apr 9 at 8:37
  • $\begingroup$ @DietrichBurde - what do you mean "See above"? $\endgroup$ Commented Apr 9 at 8:51
  • $\begingroup$ I mean, see above (first comment). So really "above". $\endgroup$ Commented Apr 9 at 9:12

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Generally if polynomials are written out, then they are usually (but not always) shown most significant term first, such as $a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0$. If shown as a summation, then least significant term first: $Σ_{i=0}^{i=n} a_i x^i = a_0 + a_1 x + ... + a_n x^n$.

Actual implementations may also operate on either least or most significant term first. For example the BCH | Reed Solomon Berlekamp Massey decoder produces polynomials of increasing degree, so some implementations operate least significant term first, while BCH | Reed Solomon Sugiyama decoder produces polynomials of decreasing degree, and most implementations would be most significant term first.

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