Overview

FEC (Forward Error Correction) adds redundant data so that the receiver can detect and correct errors without requesting retransmission. The sender computes extra parity bytes from the original data using a mathematical code, and appends them. The receiver uses the parity to detect which bytes were corrupted and compute their original values. The three FEC schemes offer different trade-offs between correction capability, complexity, and latency. Only one is used per link.

Reed-Solomon RS(255,223) (131.0-B-5)

The CCSDS standard Reed-Solomon code operates over a Galois field — a finite set of 256 values ($2^8$, one per byte) with specially defined addition and multiplication that ensure every non-zero value has an inverse. This algebraic structure is what makes the error correction mathematics work. The code appends 32 parity bytes to 223 data bytes, producing a 255-byte codeword. It can correct up to 16 corrupted bytes per codeword. In Reed-Solomon terminology, each byte is called a "symbol", and the key property is that it does not matter how badly a byte is damaged — whether one bit is flipped or all eight, it counts as a single symbol error.

For burst error resilience, the standard supports interleaving depths $I = 1$ to $5$. With interleaving, $I$ codewords are symbol-interleaved into a single block of $I \times 255$ bytes. A burst of errors that would overwhelm a single codeword is spread across $I$ codewords, each of which sees only a fraction of the damage. At depth 5 the code tolerates bursts of up to 80 consecutive corrupted bytes.

Encoding uses systematic polynomial division: the original data bytes are preserved unchanged in the output, and the 32 parity bytes are appended at the end. (A non-systematic code would intermix data and parity, making it harder to extract the data.) The parity is computed by treating the data as a polynomial over the Galois field and dividing by the generator polynomial $g(x) = \prod_{i=0}^{31} (x - \alpha^{112+i})$; the remainder of this division becomes the 32 parity bytes. Decoding uses the Berlekamp--Massey algorithm to find the error locator polynomial, Chien search to find error positions, and the Forney algorithm to compute error magnitudes.

RS is the most widely deployed FEC in space communications. It is computationally inexpensive, has deterministic decoding latency, and its byte-level correction is well-suited to the symbol-level errors that occur after demodulation.

See the detailed Reed-Solomon page for the full mathematical treatment and end-to-end example.

LDPC — AR4JA (131.0-B-5)

CCSDS specifies a family of Accumulate Repeat-by-4 Jagged Accumulate (AR4JA) LDPC codes at six code rates: 1/2, 2/3, 4/5, and 7/8 at three information block sizes (1024, 4096, 16384 bits). LDPC codes achieve error correction performance close to the Shannon limit — the theoretical maximum rate at which information can be transmitted over a noisy channel with arbitrarily low error rate. In practice, this means LDPC can operate at lower signal-to-noise ratios than RS for the same error rate.

Encoding multiplies the information bits by a sparse generator matrix. Decoding uses iterative belief propagation on the parity-check matrix: each bit node and check node exchange messages about the likelihood of each bit being 0 or 1, refining the estimates over multiple iterations until convergence. The decoder accepts soft-decision input (LLRs from the demodulator), which provides several dB (decibels — a logarithmic measure of signal power ratio) of additional gain compared to hard-decision decoding where each bit is simply 0 or 1 with no confidence information.

LDPC is preferred when the link budget is tight — for example, on long-range ISL links or during periods of high atmospheric attenuation. The trade-off is higher computational cost and variable decoding latency.

See the detailed LDPC page for the full mathematical treatment and end-to-end example.

Convolutional Code (131.0-B-5)

See the Convolutional page for the full description.