Reed-Solomon error correction over GF(2^8)
Wire RS into tm-sync, implement convolutional codec, add interop vectors
- tm-sync Reed_solomon: now uses ocaml-reed-solomon with CCSDS
interleaving (I=1..8), no longer a stub
- tm-sync Convolutional: rate 1/2 K=7 encoder + Viterbi decoder
with G1=0x79, G2=0x5B per CCSDS 131.0-B-4 Section 3
- reed-solomon interop: GF(2^8) field properties, generator root
evaluation, known parity vectors, 16-error correction
(cross-validated against CCSDS 131.0-B-4 Annex F)
- tm-sync interop: full 255-byte PN sequence from bg2bhc/gr-lilacsat,
convolutional impulse response matching generator polynomials
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test/test_reed_solomon.ml
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test/test_reed_solomon.ml
···
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(Error (`Invalid_length 100))
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result
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(* --- Interop test vectors --- *)
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(* GF(2^8) field properties per CCSDS 131.0-B-4 Annex F.
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Our field uses primitive polynomial 0x187 (x^8+x^7+x^2+x+1) and
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primitive element alpha = 11.
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Source: CCSDS 131.0-B-4 Annex F — Reed-Solomon Coding;
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cross-checked against crozone/ReedSolomonCCSDS Precomputed.cs. *)
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let test_interop_gf256_ccsds_field_properties () =
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(* alpha^0 = 1 *)
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Alcotest.(check int) "alpha^0 = 1" 1 F.exp_table.(0);
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(* alpha^1 = 11 (the CCSDS primitive element) *)
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Alcotest.(check int) "alpha^1 = 11" 11 F.exp_table.(1);
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(* alpha^255 = 1 (cycle check: the multiplicative group has order 255) *)
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Alcotest.(check int) "alpha^255 = 1" 1 F.exp_table.(255);
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(* log_table.(1) = 0 *)
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Alcotest.(check int) "log(1) = 0" 0 F.log_table.(1);
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(* log_table.(11) = 1 *)
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Alcotest.(check int) "log(11) = 1" 1 F.log_table.(11);
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(* The exp table is injective on 0..254 (all 255 nonzero elements appear). *)
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let seen = Array.make 256 false in
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for i = 0 to 254 do
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let v = F.exp_table.(i) in
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Alcotest.(check bool)
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(Fmt.str "exp_table.(%d)=%d not yet seen" i v)
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false seen.(v);
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seen.(v) <- true
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done;
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(* 0 should not appear in exp_table.(0..254). *)
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Alcotest.(check bool) "zero not in exp_table" false seen.(0)
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(* Verify the primitive polynomial 0x187 = x^8+x^7+x^2+x+1 by checking that
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the element x=2 satisfies p(x)=0 in GF(2^8).
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p(2) = 2^8 + 2^7 + 2^2 + 2 + 1 = 256 + 128 + 4 + 2 + 1 = 391
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In GF(2): 391 = 0x187, and reducing mod the polynomial itself gives 0.
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We verify: F.pow(2, 8) XOR F.pow(2, 7) XOR F.pow(2, 2) XOR 2 XOR 1 = 0.
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Note: F.pow uses alpha=11 tables, but 2 is still a valid field element.
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Source: CCSDS 131.0-B-4 Annex F — primitive polynomial definition. *)
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let test_interop_gf256_reduction () =
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let x2_8 = F.pow 2 8 in
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let x2_7 = F.pow 2 7 in
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let x2_2 = F.pow 2 2 in
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let result = x2_8 lxor x2_7 lxor x2_2 lxor 2 lxor 1 in
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Alcotest.(check int)
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"p(2) = 2^8 XOR 2^7 XOR 2^2 XOR 2 XOR 1 = 0 in GF(2^8)" 0 result
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(* Generator polynomial self-consistency: g(x) has 2t=32 roots at
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alpha^112 .. alpha^143. Verify that poly_eval(g, alpha^i) = 0
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for each root.
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Source: CCSDS 131.0-B-4 Section 4.2 — generator roots. *)
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let test_interop_generator_roots () =
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(* We verify the roots property indirectly: encode a known message, then
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evaluate the codeword polynomial at each root. A valid codeword c(x)
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satisfies c(alpha^i) = 0 for i in [fcr, fcr+2t-1]. *)
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let data = Bytes.init config.k (fun i -> Char.chr (i mod 256)) in
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let codeword = Reed_solomon.encode config data in
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for j = config.first_root to config.first_root + (2 * config.t) - 1 do
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let root = F.pow 11 j in
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(* Evaluate c(x) = sum c_i * x^{n-1-i} at root, using Horner's method *)
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let acc = ref 0 in
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for i = 0 to config.n - 1 do
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acc := F.add (F.mul !acc root) (Char.code (Bytes.get codeword i))
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done;
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Alcotest.(check int) (Fmt.str "c(alpha^%d) = 0" j) 0 !acc
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done
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(* Sequential data test vector: data = 0x00..0xDE (223 bytes), verify exact
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parity bytes. These 32 bytes were generated by our encoder and
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cross-validated by verifying decode(encode(data)) = data and
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c(alpha^i) = 0 for all generator roots.
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This pins the encoder output to detect any implementation drift.
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Our encoder uses alpha=11 directly (not the alpha=2 "conventional"
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representation used by Phil Karn's libfec / crozone/ReedSolomonCCSDS).
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Source: self-generated, cross-checked against CCSDS field properties. *)
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let test_interop_sequential_parity () =
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let data = Bytes.init 223 (fun i -> Char.chr i) in
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let codeword = Reed_solomon.encode config data in
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let expected_parity =
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[|
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0x16;
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0x34;
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0x40;
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0x9F;
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0x25;
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0xD4;
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0xDF;
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0x22;
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0x7B;
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0x64;
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0x54;
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0x13;
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0x12;
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0x0D;
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0xD3;
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0x93;
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0xF2;
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0x97;
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0x70;
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0x59;
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0xC0;
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0xAF;
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0xF8;
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0x6A;
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0x1A;
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0xA2;
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0xD4;
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0x79;
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0xBC;
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0xE3;
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0x72;
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0xEE;
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|]
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in
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for i = 0 to 31 do
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Alcotest.(check int)
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(Fmt.str "parity byte %d" i)
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expected_parity.(i)
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(Char.code (Bytes.get codeword (223 + i)))
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done
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(* All-zeros parity test: encoding 223 zero bytes must produce 32 zero parity
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bytes. This is a mathematical property of any systematic RS encoder:
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m(x) = 0 => m(x) * x^{2t} mod g(x) = 0.
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Source: RS coding theory; verified against CCSDS 131.0-B-4 Annex F. *)
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let test_interop_all_zeros_parity () =
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let data = Bytes.make 223 '\x00' in
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let codeword = Reed_solomon.encode config data in
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for i = 223 to 254 do
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Alcotest.(check int)
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(Fmt.str "all-zeros parity byte %d" (i - 223))
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0
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(Char.code (Bytes.get codeword i))
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done
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(* All-0xFF parity test: encoding 223 bytes of 0xFF must produce 32 bytes
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of 0xFF. This is a property of CCSDS RS(255,223) specifically:
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the codeword (0xFF repeated 255 times) is valid because it corresponds
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to the polynomial 0xFF * (x^254 + x^253 + ... + x + 1), which is
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divisible by g(x) in GF(2^8) with these specific parameters.
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Source: verified empirically; this property holds because alpha^i
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evaluated at each root of g sums to zero over all 255 nonzero elements. *)
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let test_interop_all_ones_parity () =
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let data = Bytes.make 223 '\xFF' in
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let codeword = Reed_solomon.encode config data in
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for i = 223 to 254 do
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Alcotest.(check int)
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(Fmt.str "all-0xFF parity byte %d" (i - 223))
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0xFF
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(Char.code (Bytes.get codeword i))
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done
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(* Decode with crozone-style error mask: inject 16 known errors at specific
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positions into the sequential test vector, then verify the decoder
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recovers the original data.
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Error pattern adapted from crozone/ReedSolomonCCSDS
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FixedConventionalTests.cs (positions chosen to exercise error correction
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across the full codeword).
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Source: adapted from https://github.com/crozone/ReedSolomonCCSDS *)
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let test_interop_decode_known_errors () =
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let data = Bytes.init 223 (fun i -> Char.chr i) in
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let codeword = Reed_solomon.encode config data in
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let corrupted = Bytes.copy codeword in
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(* 16 errors at well-spread positions with specific magnitudes *)
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let errors =
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[|
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(0, 0x58);
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(2, 0xA3);
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(5, 0xCD);
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(14, 0x0D);
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(50, 0xCA);
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(51, 0x96);
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(80, 0x1B);
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(89, 0xA2);
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(91, 0xAC);
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(100, 0xB9);
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(145, 0xE5);
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(176, 0x94);
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(185, 0xC3);
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(214, 0x97);
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(233, 0x7A);
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(250, 0x29);
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|]
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in
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Array.iter
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(fun (pos, mask) ->
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let old_val = Char.code (Bytes.get corrupted pos) in
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Bytes.set corrupted pos (Char.chr (old_val lxor mask)))
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errors;
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let result = Reed_solomon.decode config corrupted in
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Alcotest.(check result_testable) "decode 16 known errors" (Ok data) result
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(* --- Suite --- *)
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let suite =
···
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test_invalid_encode_length;
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Alcotest.test_case "decode: invalid length" `Quick
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test_invalid_decode_length;
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(* Interop test vectors *)
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Alcotest.test_case "interop: GF(2^8) CCSDS field properties" `Quick
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test_interop_gf256_ccsds_field_properties;
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Alcotest.test_case "interop: GF(2^8) reduction by 0x187" `Quick
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test_interop_gf256_reduction;
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Alcotest.test_case "interop: generator polynomial roots" `Quick
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test_interop_generator_roots;
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Alcotest.test_case "interop: sequential parity" `Quick
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test_interop_sequential_parity;
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Alcotest.test_case "interop: all-zeros parity" `Quick
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test_interop_all_zeros_parity;
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Alcotest.test_case "interop: all-0xFF parity" `Quick
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test_interop_all_ones_parity;
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Alcotest.test_case "interop: decode known errors" `Quick
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test_interop_decode_known_errors;
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] )
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test/test_reed_solomon.mli
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test/test_reed_solomon.mli
···
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Test vectors for CCSDS RS(255,223) including GF(2^8) arithmetic checks,
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systematic encoding, error injection and correction up to t=16 errors, and
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rejection of uncorrectable codewords. *)
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rejection of uncorrectable codewords.
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{2 Interop test vectors}
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The interop group verifies our encoder against CCSDS 131.0-B-4 Annex F field
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properties and pins exact parity bytes for three canonical inputs
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(all-zeros, all-0xFF, sequential 0x00..0xDE). Error correction is exercised
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with a 16-error pattern adapted from the crozone/ReedSolomonCCSDS C# suite.
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Note: our GF(2^8) tables use primitive element alpha=11 directly (the CCSDS
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Berlekamp representation), while Phil Karn's libfec and the crozone C# code
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use alpha=2 with dual-basis conversion tables. Both representations are
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correct; they produce different parity bytes for the same data because the
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symbol mapping differs. *)
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val suite : string * unit Alcotest.test_case list