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I have code encoded in GF(7) with primitive 5 Сf(4,1,0,4,5,5). (last four symbols is redundancy)

While decoding using DFT we use formula $$ С_k=N^{-1}*c(z^{-kj}) $$

example:

$$ C_1 = c(5^{-1*j})/6 = (4*5^{-0*1} + 1*5^{-1*1} + 0*5^{-2*1} + 4*5^{-3*1} + 5*5^{-4*1} + 5*5^{-5*1})/6 = (4 + 3/15 + 24/750 + 20/2500 + 25/15625)/6 = (4 + 3 + 24 + 20 + 25)/6 = 76/6 = 456/36 = 456 = 1; $$ $$ C_2 = c(5^{-2*j})/6 = (4*5^{-0*2} + 1*5^{-1*2} + 0*5^{-2*2} + 4*5^{-3*2} + 5*5^{-4*2} + 5*5^{-5*2})/6 = (4 + 2 + 4 + 10 + 20)/6 = 40/6 = 240/36 = 240 = 2; $$

I don't understand how to work with negative powers.

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  • $\begingroup$ Use $5^6\pmod 7 =1,$ to re-express negative powers of $5$ with exponents in $\{0,1,\cdots,5\}.$ $\endgroup$ – kodlu Mar 26 at 14:04
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Use $5^6\pmod 7 =1,$ to re-express negative powers of $5$ with exponents in $\{0,1,\cdots,5\}.$ This works since $5$ is a generator of the multiplicative group of $GF(7).$

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