The Distance Spectrum of IEEE 802.11 Binary Convolutional Codes

TL;DR

This note explains how to compute the distance spectrum of the convolutional code exactly for the IEEE 802.11 mother code and its three standard punctured derivatives and its three standard punctured derivatives obtained via rate-compatible puncturing.

Abstract

Binary convolutional coding (BCC) has been a cornerstone of the IEEE 802.11 wireless LAN standard since its inception, and it remains relevant today across the full generational arc from the legacy 802.11a/g through Wi-Fi 6 (802.11ax) and into the forthcoming Wi-Fi 8 (802.11bn). Although low-density parity-check (LDPC) codes now dominate high-throughput applications, BCC is mandatory for backward compatibility and continues to serve as the default forward-error-correction scheme in bandwidth-constrained and cost-sensitive deployments: 20 MHz-only devices, Internet-of-Things nodes, and other implementations where LDPC's decoder complexity is prohibitive. Critically, BCC at rate 1/2 is the coding scheme used throughout the packet preamble in every IEEE 802.11-compliant frame, making it indispensable regardless of which data-field code is selected. Furthermore, the new Enhanced Long Range (ELR) packet format introduced in the 802.11bn/UHR amendment mandates rate-1/2 BCC for the data portion of the frame, reinforcing the continued importance of this code in next-generation deployments. The performance of BCC under Viterbi decoding is governed by the distance spectrum of the convolutional code. This note explains how to compute that spectrum exactly for the IEEE 802.11 mother code (rate 1/2, K=7, generators octal 133 / octal 171) and its three standard punctured derivatives (rates 2/3, 3/4, 5/6) obtained via rate-compatible puncturing. Union-bound BEP and FER curves are derived for AWGN with BPSK/QPSK and Gray-coded M-QAM modulation and validated against Monte Carlo simulation. Python, Julia, and C++ implementations are openly available at https://github.com/geekymode/bcc_spectrum.

Figures (8)

• Figure 1: IEEE 802.11 $K=7$ BCC shift-register encoder. Each box labelled $D$ is a unit-delay element; $\oplus$ denotes mod-2 addition. The current input $u_n$ enters at the filled dot (tap position 0, no delay); the six $D$ elements hold $u_{n-1},\ldots,u_{n-6}$. The $g_1=133_8$ XOR chain runs above (taps at positions 0, 2, 3, 5, 6); the $g_2=171_8$ chain runs below (taps at positions 0, 1, 2, 3, 6). All tap lines are purely vertical, so no wire crosses any bus segment.
• Figure 2: Rate-$\frac{1}{2}$ trellis, 4 lowest states over $n=0,\ldots,7$. Blue solid: $u=0$; red dashed: $u=1$. Edge labels $(v^{(1)}v^{(2)})$ are shown for the first transition only; the pattern repeats identically at every subsequent step. The full code has 64 states; here $\sigma\in\{00,01,10,11\}_2$.
• Figure 3: 802.11 puncture masks and free distances.
• Figure 4: Rate-$\frac{3}{4}$ mask 111001 over two periods. Solid border: transmitted. Dashed border: deleted before transmission. Superscript indexes the generator ($1$ or $2$); subscript indexes the input-bit time step. 4 of every 6 serial output bits are transmitted, giving rate $\frac{4/2}{6/2}=\frac{2}{3}\cdot\frac{3}{2}=\frac{3}{4}$.
• Figure 5: BEP union bound \ref{['eq:ub']} (left) and FER union bound \ref{['eq:fer']} (right, $K=1024$ information bits) for all four IEEE 802.11 BCC rates over AWGN with QPSK modulation proakis01viterbi67. Lines: union bounds using 30 spectrum terms ($d_{\max}=130$), solid for both BEP and FER. Markers: Monte Carlo simulation, dot-dashed lines; $\bullet$$R=\frac{1}{2}$, $\blacksquare$$R=\frac{2}{3}$, $\blacktriangle$$R=\frac{3}{4}$, $\blacklozenge$$R=\frac{5}{6}$. Dotted black: uncoded BPSK $Q\!\left(\sqrt{2E_b/N_0}\right)$ (BEP panel only). The FER bound lies roughly $K/\bar{w}$ times above the BEP bound ($\bar{w}\approx3$--$7$ per rate); both bounds track the simulation closely in the waterfall region.

Theorems & Definitions (2)

• Definition 1: Distance spectrum lin04ryan09viterbi79
• Definition 2: Augmented state