added Cauchy Reed Solomon erasure coding

Read more about the Cauchy matrix:

https://en.wikipedia.org/wiki/Cauchy_matrix

The Art of Computer Programming
Volume 1: Fundamental Algorithms
Donald E. Knuth - 1997
Paragraph 1.2.3: Sums and Products

Page 37, Cauchy's matrix:
$a_{ij} = \frac{1}{x_i + y_j}$

Page 38, Exercise 41:
$b_{ij} = \frac{\prod_{k=1}^{n}{(x_j + y_k)(x_k + y_i)}}{(x_j + y_i)\prod_{k \ne j}^{n}{(x_j - x_k)}\prod_{k \ne i}^{n}{(y_i - y_k)}}$

tests/crs_regression_test.cc

@@ -0,0 +1,70 @@
+/*
+Regression Test for the Cauchy Reed Solomon Encoder and Decoder
+
+Copyright 2023 Ahmet Inan <inan@aicodix.de>
+*/
+
+#include <cassert>
+#include <random>
+#include <iostream>
+#include <functional>
+#include "galois_field.hh"
+#include "cauchy_reed_solomon_erasure_coding.hh"
+
+template <typename GF>
+void crs_test(int trials)
+{
+	CODE::CauchyReedSolomonEncoder<GF> encode;
+	CODE::CauchyReedSolomonDecoder<GF> decode;
+	std::random_device rd;
+	std::default_random_engine generator(rd());
+	typedef std::uniform_int_distribution<int> distribution;
+	auto rnd_cnt = std::bind(distribution(1, std::min(GF::Q / 2, 256)), generator);
+	auto rnd_len = std::bind(distribution(1, 1 << 10), generator);
+	auto rnd_dat = std::bind(distribution(0, 255), generator);
+	while (--trials) {
+		int block_count = rnd_cnt();
+		int numbers_total = GF::Q - block_count;
+		int block_bytes = rnd_len() * sizeof(typename GF::value_type);
+		int data_bytes = block_count * block_bytes;
+		uint8_t *orig = new uint8_t[data_bytes];
+		for (int i = 0; i < data_bytes; ++i)
+			orig[i] = rnd_dat();
+		uint8_t *blocks = new uint8_t[data_bytes];
+		auto numbers = new typename GF::value_type[numbers_total];
+		for (int i = 0; i < numbers_total; ++i)
+			numbers[i] = i;
+		for (int i = 0; i < block_count; i++) {
+			auto hat = std::bind(distribution(i, numbers_total - 1), generator);
+			std::swap(numbers[i], numbers[hat()]);
+		}
+		for (int i = 0; i < block_count; ++i)
+			encode(orig, blocks + block_bytes * i, numbers[i], block_bytes, block_count);
+		uint8_t *data = new uint8_t[data_bytes];
+		decode(data, blocks, numbers, block_bytes, block_count);
+		for (int i = 0; i < data_bytes; ++i)
+			assert(data[i] == orig[i]);
+		delete[] numbers;
+		delete[] blocks;
+		delete[] orig;
+		delete[] data;
+	}
+}
+
+int main()
+{
+	if (1) {
+		typedef CODE::GaloisField<8, 0b100011101, uint8_t> GF;
+		GF instance;
+		crs_test<GF>(200);
+	}
+	if (1) {
+		typedef CODE::GaloisField<16, 0b10001000000001011, uint16_t> GF;
+		GF *instance = new GF();
+		crs_test<GF>(100);
+		delete instance;
+	}
+	std::cerr << "Cauchy Reed Solomon regression test passed!" << std::endl;
+	return 0;
+}
+