Suppose I create a polynomial ring:
F.a = GF(2^4) R.x = PolynomialRing(F)
The syntax above is slightly incorrect, I couldn't figure out how to preserve the brackets around the 'a' and 'x'.
For the work I'm doing the references typically refer to elements of the ring in 4 different manners (within the same reference). An example of this is the BBC white paper, "Reed-Solomon Error Correction" by C. K. P. Clarke.
They use index form:
a^11
They use polynomial form:
a^3 + a^2 + a
They use binary form:
1110
They use decimal form:
14
I would like to map what they call decimal form to the polynomial form, because it's very convenient. For instance in the previously mentioned paper they encode a message: x^10 + 2x^9 + 3x^9 + ... 10*x + 11.
In sagemath this would be:
x^10 + ax^9 + a^4x^8 + ... a^9*x + a^7
The decimal form is very helpful for me. My eventual goal is to use sagemath to implement a Reed-Solomon encoder/decoder, and write code that parses that implementation and creates System Verilog code that gets implemented in an ASIC (application specific integrated circuit)
