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Category Archives: Math
Evaluating Polynomials in an FPGA
Sometimes, a polynomial will need to be evaluated at a specific value in an algorithm. This might be used for curve fitting or interpolation. Another case is for Galois Fields and error correction, where high-degree polynomials are common. There is … Continue reading
Another Galois Field Multiply
After completing my Reed-Solomon decoder’s HDL, I decided to see how well it performed. The results weren’t as good as I expected. The limiting path was in the field math. A bit of simple pipelining allowed 250MHz operation on my … Continue reading
Posted in FPGA, Math
Tagged code provided, galois field
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The 360 Element Shift Register
As mentioned in the three-bit shift register article, one of the applications for non-maximal length LFSRs/NLFSRs is in angle sensors. I had read an article on generating LFSRs of a given length. For the 360 element case, it came up … Continue reading
Reed-Solomon Encoder
This article describes how to make a performance optimized Reed-Solomon encoder. The actual encoder is for an full-length, narrow-sense RS code. In this case, the code generated was an code that could allow at least 8 bytes to be corrected, … Continue reading
Three-Bit Feedback Shift Register
One question that I’ve often pondered is how to generate non-maximal length LFSRs or NLFSRs. I decided to see what could be done with an 8b shift register that took 3 taps, and performed a logic operation to generate the … Continue reading
Shifting an LFSR Multiple Times
One useful operation for a Galois Field is the multiplication by a constant value. This is very useful, as it allows very simple logic to be used to generate the product . This avoids the need to convert between polynomial … Continue reading
Basic Math In a Galois Field
A Galois Field presents unique challenges, as basic operations like addition and multiplication are different. This article addresses basic methods for addition, subtraction, multiplication, and division of field elements. FPGA considerations are included.
Intro to Galois LFSRs
There is another, natural way to write the code for an LFSR. This article describes the Galois implementation of an LFSR. This also shows how LFSRs are related to Galois fields. This implementation is claimed to be a bit faster … Continue reading
The Magic LFSR Taps
Just like magic, these aren’t real either. Continue reading