03 — Coding layer (scrambler / interleaver / FEC / CRC)

Two coding regimes exist and must not be conflated:

• MFSK handshake bursts (CR / connect-response / connected-ack) carry no FEC, interleaving or whitening — their tones are a public-LCG (VB6 Rnd) PRNG stream keyed to the destination callsign, specified in `04 §4.2`. They do not use the stages in this file. The one shared piece is the CRC (§3.4), which derives the PRNG seed.
• DATA levels of every family and bandwidth (index, tone and OFDM, `01 §1.0`; host BITRATE (N) … bps): §3.0–§3.4 apply to all of them. §3.5 is the BW2300 index levels and the link-setup, §3.6 the BW500 base path, §3.7 the tone and OFDM levels.

3.0 Pipeline order

One chain serves every DATA level; only the last stage, bits to cells, belongs to the waveform family.

frame    payload | control byte | CRC-16/GENIBUS(payload | control), big-endian
bits     MSB first, zero-padded to the record's N info bits
whiten   XOR the PN (§3.1) from index 0
turbo    (13,15) on the record's stage-2 permutation (§3.2, §3.3)
puncture group law to the record's coded length (§3.3); none at rate 1/2 or 1/3
gather   onair[k] = coded[π1[k]], π1 the record's stage-1 permutation
cells    index: bpc bits → Gray → one lit bin ([01 §2300])
         tone:  2 bits → Gray quarter-turn increment per band ([01 §Tone family])
         OFDM:  bpc bits → LUT column pos, then × exp(−j2π·S/128) ([01 §OFDM family])

RX is the exact inverse. The MFSK handshake bursts have no stage of this chain (see `04 §4.2`).

3.1 Scrambler / randomizer (DATA)

The OFDM levels add a second, cell-domain scramble after modulation, drawn from the same seed (§3.7).

3.2 Interleaver (DATA)

3.3 Forward error correction (FEC) (DATA)

Group puncture. Wherever coded < 3N + 12:

q = coded − N − 12,  k = round(N / q)
group g carries info bits s[kg .. kg+k−1] as   s[kg], P_g, s[kg+1], …, s[kg+k−1]
P_g = p1[kg] for even g, p2[kg] for odd g      (constituent 1 / 2 parity at the group's first bit)
then the 12-bit dual tail, then zeros up to coded

k = 1 is the rate-1/2 mux (s p1 s p2 …); rate 2/3 reads s p1 s | s p2 s. coding/turbo.py encode_punctured / decode_punctured. Per OFDM level (the tone rows are all k = 1, pad 0):

3.4 CRC / error detection

3.5 BW2300 index levels and the link-setup

The BW2300 index levels carry both the caller-ID link-setup burst and the base level's data. §3.5.3 gives the chain and its framing, §3.5.4 the constant tables, §3.5.5 the caller-ID link-setup frame.

3.5.3 BW2300 index-level link-setup / DATA chain

The BW2300 index levels (host L1–4, records 0–3), the link-setup included, use the §3.0 chain with the index-family cell stage. In functional terms (the bin law is in `01 §2300`):

DEMOD:  per emission column, the lit bin → Gray⁻¹ → bpc on-air bits
        → channel de-interleave (the record's stage-1 column, gather)
        → turbo (13,15) log-MAP decode (the record's stage-2 column)
        → de-whiten (XOR the PN, §3.1)
        → CRC-16/GENIBUS check  → frame → caller callsign + session params
MOD  =  the exact inverse (forward) chain.

The modulation is index/position type — the coded value selects the carrier bin — so RX is an argmax over the received energy grid.

• Framing. A frame is 92 bytes = 736 bits: [89 payload bytes][1 control byte][2-byte CRC, big-endian], with no header. CRC-16/GENIBUS over body[0:90] equals frame[90:92] big-endian. The link-setup burst uses the same 92-byte framing law as a DATA over.
• Per-over payload slicing. 89 consecutive payload bytes per over (over_i = payload[89·i : 89·i+89]); a 512-byte payload therefore occupies 6 overs.
• Whitener. XOR with the PN of §3.1, MSB-first, from index 0 per frame.
• Turbo. Parallel (13,15)-octal RSC, rate 1/2, N = 736 → 1484 coded bits (= 2·736 + 12 dual-tail) at records 3 and 2, and rate 1/3, `3N + 12` at records 1 and 0 — 612 from 200 and 300 from 96. The internal interleaver is interleave_stage2.i32 column 3, perm[i] = IL2[3 + 17·i][:736] — a native 0..735 permutation, used directly with no pruning.
• Channel interleaver. interleave_stage1.i32 column 3 (a 1484-entry permutation), applied as a GATHER: onair[k] = coded[π[k]].

3.5.4 BW2300 coding data tables

The BW2300 constant tables needed to build a VARA-accurate BW2300 link-setup / DATA transmitter are produced in code by the companion implementation's generator, hfmodem/kestrel/rx/tablegen.py, under a byte-exactness gate; they are not shipped as data files, and the file names in the table below identify the tables the generator produces. The generator records how each is known. The forward (MOD) chain is fully specified by these tables plus the `01 §2300` raster.

3.5.5 Caller-ID link-setup coding

Reading the burst. It is easy to take this burst for a narrowband 16-ary MFSK waveform with 1024-sample symbols. It is the ordinary BW2300 rec3 wideband OFDM burst: the apparent "16 tones, one tone per symbol" is rec3's one-hot index modulation (one lit bin in a 16-bin span, bpc = 4), the apparent "371 symbols" are the 371 data emission columns, and symbols are 512 samples. The coding below is independent of that geometry. See `04 §4.2A`.

The caller-identity link-setup burst is coded with the same (13,15)-octal turbo family as the BW500 DATA path (§3.6.4), configured rate 1/2, N = 736:

TX:  caller frame (90 body bytes + 2-byte CRC-16/GENIBUS BE, big-endian, over body[0:90])
     → serialize to 736 bits (MSB-first) → XOR PN whitener (BW2300 whitener PN)
     → turbo (13,15) rate-1/2 encode (+12-bit dual tail)      → 1484 coded bits (encoder-native)
     → CHANNEL interleave (GATHER): onair[k] = coded[ π[k] ], π = interleave_stage1 col-3 [:1484]
     → group 4 on-air bits MSB-first → Gray σ symbol map (LUT [0,1,3,2,7,6,4,5,15,14,12,13,8,9,11,10])
     → 371 rec3 data emission columns (Gray value → one-hot bin, per [`01 §2300`](01-physical-layer.md))
RX = the inverse: σ⁻¹ demap → channel de-interleave (coded[π[k]]=onair[k]) → turbo log-MAP
     decode → de-whiten → CRC-16/GENIBUS → frame.

N-parametric rule (both interleavers). Both permutations come from VARA's two 17-column int32 tables, accessed as perm[i] = TABLE[i·17 + col]: channel/carrier-placement from interleave_stage1.i32 (column chosen per *coded* length; col-3 → 1484), and internal turbo from interleave_stage2.i32 (column chosen per *info* block size). A column is used directly when its first-N entries already form a 0..N-1 permutation. Every level's column does, including the caller-ID N=736 case, so the column is always taken directly: perm[i] = TABLE[i·17 + level][:N]. Do not derive the permutation by pruning a larger column — dropping entries ≥ N from the next larger column, e.g. building N=736 out of the N=744 column. That yields a different permutation, and a modem built to it will not interoperate.

3.6 BW500 DATA-path coding

This section specifies the BW500 DATA path so it can be built from spec alone: level 4 (SHORT, record 3) in §3.6.1–§3.6.5 and the level below it (record 2) in §3.6.6, plus the rate-1/3 variant used by control levels 0/1 in §3.6.4. The BW500 constant tables are produced in code by the companion implementation's generator, hfmodem/kestrel/rx/tablegen.py (§3.5.4), rather than shipped as data. Nothing ships: the tables that follow from the algorithms below are computed at the point of use, and the two the algorithms do not reach — the measured de-rotation integers and BW2300 record 2's base bins — are integer literals in the same module.

3.6.1 Physical / demod frame (2-subband, L4 SHORT)

Analysed as a single band on a 256-sample symbol, the 500 Hz waveform does not resolve into a recoverable constellation. It is two sub-bands at H = 512, as above and in `01 §500`.

3.6.2 De-rotation reference + cell selection

3.6.3 Interleavers (Stage-1 raster + turbo interleaver)

3.6.4 Whitener + turbo FEC + CRC

3.6.5 TX/RX pipeline order (BW500 L4)

TX:  frame(43B + control byte) → CRC-16/GENIBUS append (2B, big-endian)
     → serialize to 368 bits → XOR PN whitener
     → turbo (13,15) rate-1/2 encode (+12-bit dual-tail)   → 748 coded bits
     → Stage-1 raster place  P = IDX1[3::17][:748]
     → constellation/diff-BPSK map onto data cells (usedmap-selected)
     → ⊗ grid480 reference (twiddle)  → 2-subband WOLA synth (H=512, 1350/1650 Hz)
RX = exact inverse (de-rotate → diff-BPSK → Stage-1 de-interleave
     → turbo log-MAP decode → de-whiten → CRC check).

3.6.6 Record 2 (host BITRATE (3)) — one-hot index modulation

The gear below the base level, 61 bps, and the level a stock BW500 pair opens a delivery at. It is not §3.6.1's waveform slowed down: it is one-hot index modulation, the family BW2300 records 2 and 3 use (§3.5.4), on this bandwidth's own tables.

Known from the two record-2 overs of the 2026-07-13 bw500-zeros256 loopback session: every one of the 226 columns of both overs lights the bin these tables predict, and both decode CRC-clean to the 34 and 7 payload bytes the session's host log records. Reference columns 73 and 74 satisfy every constraint two overs can raise; the closed form above takes 74, and either choice decodes both overs to the same bytes.

The 4 training columns are open. They differ between the two overs, so they carry a per-over draw, and no Rnd stream anchored anywhere reproduces both. The wide bandwidth's law — the record's own alloc read from index 1, offset by Int(Rnd·16) per column (§3.5.4) — has no solution here at the gap that law is defined with, and the two overs' eight bins cannot fix a 24-bit stream position and a gap besides: 32 of 40 invented bin pairs admit a solution on the same search. BW500 offers no anchor either, its base level being sub-band DBPSK with no one-hot preamble to invert (best single-bin share of a base lead-in, over every phase and both combs: 0.32, against 1.00 for these columns).

Nothing reads them for correctness — a receiver cannot know the transmitter's stream position, the two ends of a link sitting at different states — so they are training: energy in the comb for gain, timing and frequency. A transmitter keys the allocation with the draw at zero, and the 226 frame columns behind it are exact: rendered against the recording, the body correlates 0.99997 and the whole over 0.987, the difference being those four columns alone.

3.7 Tone and OFDM levels

The coding half of the tone family (BW2300 L5–9, BW2750 L5–10, BW500 L5–6) and the OFDM family (BW2300 L10–16, BW2750 L11–17, BW500 L7–13). Their physical half is `01 §1.0` and the sections it points to; per-level N, coded and frame sizes are `06`. Closed from known plaintext on the high speed capable bench tapes: every frame of every row decodes CRC-clean and byte-exact against the transcript (evidence/KPA-bw2300-const.md, KPA-bw2300-subband.md, KPA-bw2750-const.md, KPA-bw500-const.md, KPA-tone-bw2750-bw500.md).

Frame. floor(N/8) bytes: payload, one control byte, then CRC-16/GENIBUS over both, big-endian; N bits are the frame zero-padded where N is not a byte multiple (2/4/2 bits at BW2300 L6/7/8). The control byte is the ARQ layer's per-frame field (`04 §4.2B`): 0x81 to 0x99 in steps of 4 on full frames, 0x19 on a session's first data over, 0x82 on a short one. A payload short of the row's capacity is filled ahead of the control byte:

spare ≥ 3   payload | 14 <caller CRC high> | 00 … | T | 82 | CRC
spare = 2   payload | 14 T | 82 | CRC
T = 06 at BW2300 and BW2750, 07 at BW500

<caller CRC high> is the high byte of CRC-16/GENIBUS of the session caller's callsign, as in every short over (`04 §4.2B`); the high speed capable tapes carry 09 (N0CALL calling) and f6 (W9SSJ).

Chain. §3.0 unchanged: whiten, turbo on the record's stage-2 permutation, group puncture where the rate is above 1/2 (§3.3; every tone row is rate 1/2), gather by the stage-1 permutation. The permutations are the bandwidth chain's own records (§3.2): BW2750 L5–9 are BW2300 records 4–8 outright, and BW500 L5–6 are BW500 records 4–5.

Tone cells. Two on-air bits MSB first → Gray [0,1,3,2] → a quarter-turn increment, filled band-major (band 0's data columns in time order, then band 1's), skipping pilot columns (`01 §Tone family`).

Tone pilots (tablegen.tone_pilots). Per band in order, draw co = 5 + Int(Rnd·(ncols − 6)) from the record's gap, rejecting repeats, until the band has its count; t = 8 + co. The draws close on the next record's stage-1 seed: 37 / 36 / 36 / 40 / 41 draws at L5–9 (BW2300 and BW2750 alike), 39 at BW2750 L10, 38 and 36 at BW500 L5 and L6. BW500 record 3's 36 pilots (L4) are the same law.

OFDM cells. bpc on-air bits MSB first as v, cell = LUT[v] from LUT column pos indexed by v directly (tablegen._column(pos, v)). There is no Gray pass in front at any column: the clsparm Gray rows belong to the index levels only, and all 32 labels of the pos-4 cross are observed on tape. Fill is bin-major, symbol order within a carrier.

Scramble (tablegen.scramble()), OFDM only:

state ← 29318 (the whitener's seed)
for k in 0..61 (FFT bin), c in 0..408 (column): state ← Rnd step; S[k][c] = Int(Rnd·129)
on air: cell · exp(−j2π·S[k][c] / 128)
c = s − 4 (PSK, column 0 the reference symbol) or s − 5 − 154·f (QAM, frame f)

So bin k column c is draw 1 + 409·k + c: bin 8 column 0 is draw 3273 (BW2300), bin 3 draw 1228 (BW2750), bin 27 draw 11044 (BW500). The lead is not scrambled.

Gap cells (tablegen.gap_cells), OFDM only. From the record's gap, carrier by carrier from the first bin, draw 7 + Int(Rnd·142) (PSK: the symbol of a repeat cell, a +1 differential step) or Int(Rnd·19) (QAM: the pilot row whose next row holds an extra +1), redrawing a value its carrier already holds. One per carrier at BW2300 (49 draws) and BW2750 (59), four at BW500 (44 plus the redraws, 44–49). The draws close exactly on the next record's stage-1 seed, which is what fixes the gap as these cells.