Web Simulation 

  

 

 

 

3GPP NR Polar Coding Tutorial 

This interactive tutorial visualizes 3GPP NR Polar Coding (TS 38.212): the encoding pipeline from information bits to codeword via mother code selection, interleaving, sub-channel mapping, and the polar transform. The flow is: Information → Interleaved (c′) → U-vector (Frozen/Info) → Polar Transform → Codeword (d).

 

Sections

Mathematical foundation

1. Mother code size N

Given payload length K (information bits) and rate-matching length E, 3GPP selects the mother code size N = 2n (5 ≤ n ≤ 9). The rule uses n1 from E (and a threshold), n2 from K, and clamps to the allowed range. N is the length of the U-vector and the codeword before rate matching.

2. Interleaving (PIIL)

The block interleaver (Table 5.3.1.1-1, PIIL_MAX) shuffles the K information bits into the sequence c′. The mapping is defined by the 3GPP table: for each output position, a source index is chosen from the last K entries of the sequence. The interleaver visualization shows these "strings" from input to output.

3. Sub-channel mapping (Q0Nmax)

The reliability sequence Q0Nmax (Table 5.3.1.2-1) orders sub-channels by reliability. The K most reliable positions are used for information bits; the rest are frozen (set to 0). The heatmap shows reliability (color) and selected indices (blue border). The Master Sequence Explorer shows the full 1024-element table; Step Fwd animates each bit from the table into the u-vector.

4. Polar transform (butterfly)

The U-vector u of length N (info bits in selected positions, frozen elsewhere) is multiplied by the polar kernel matrix — the n-fold Kronecker power of the 2×2 Arikan kernel F:

d = u · GN,   GN = F⊗n,   F = [1 0; 1 1]

This is implemented as a butterfly network: at each stage, pairs of bits are combined with XOR, uj ← uj ⊕ uj+step. The diagram shows input nodes (blue = info, grey = frozen), XOR nodes, and output codeword d. The Math Inspector displays the active XOR equation for the current step.

Simulation

The interactive simulator is below. Use the controls to explore the concepts described above.

1. Mother Code 2. Interleaving 3. Sub-channel Mapping 4. Polar Transform 5. Output
Mother Code (3GPP)

E (rate match)

108

K (payload)

40

N (mother)

128 (2^7)

Code rate K/E
0.370
Reliability heatmap (Q0Nmax) — Selected info bits (blue border)
Info bit Frozen bit

Set K and E, then click Generate Random Bits & Run. Mother code size N is computed per 3GPP.

3GPP Interleaver Table (PIIL_MAX, TS 38.212 Table 5.3.1.1-1)
Interleaver mapping: input index → output position (c → c′)

Sub-channel mapping: the K most reliable positions (from Q0Nmax) carry information; others are frozen (0). Think of it as a Physical Mailbox system: Rank (reliability order) tells you which mailbox is best; you place your Info bits into the K best mailboxes and fill the rest with 0. Step Fwd animates each bit "flying" from the reliability table into its physical u-vector slot.

3GPP Master Sequence Explorer (Q0,Nmax)
Scanning Master Table for indices < 128…
Table 5.3.1.2-1 (Q0Nmax, filtered for N) — Rank = reliability order; highlight follows Step
Table has N rows (Rank 0 to N−1). Scroll down to see Info rows (last K rows).
U-vector (by u[] index)
Butterfly diagram: U-vector (blue = info, grey = frozen) → XOR stages → Codeword d
Polar Transform Evolution
Stage 7 / 7
XOR 1 / 64

Use Stage ± to select the stage; XOR ± or Step Bwd/Fwd to show each XOR (⊕) within that stage. Run animates through the XORs.

Codeword d (output)

 

Usage

Use this simulation to explore 3GPP NR Polar Coding (TS 38.212):

  1. K (information bits): Payload length (1–164). E (rate matching length): Output length after rate matching. Enter values and click Generate Random Bits & Run to compute mother code size N and run the full pipeline. The 3GPP master table (Q0Nmax) is validated (1024 unique indices) before running.
  2. Step Fwd / Step Bwd: Move through the five stages: Mother Code → Interleaving → Sub-channel Mapping → Polar Transform → Output. Click a step label to jump to that stage.
  3. Mother Code table: Shows E, K, and selected N (2n, 5 ≤ n ≤ 9). Code rate K/E: Displayed below; turns red if K/E > 1 (invalid for decoding).
  4. Reliability heatmap: Sub-channels ordered by Q0Nmax. Blue border = selected info positions; others are frozen. Hover a cell for index and rank.
  5. Interleaver (Stage 2): Bezier curves show how input bits (left) map to interleaved positions (right) per PIIL_MAX. Use Step Bwd/Fwd or Animate Interleave.
  6. Sub-channel Mapping (Stage 3): Master 3GPP Sequence Explorer ribbon shows the full Q0,Nmax table; highlight and scroll sync with the current rank. Step Fwd animates each bit from the table row into the u-vector slot; Step Bwd / Reset reverse or clear.
  7. Butterfly (Stage 4): U-vector input nodes (blue = info, grey = frozen), XOR nodes (orange), and codeword output (green). Use Stage ± and XOR ± or Step Bwd/Fwd to advance; Math Inspector shows the active XOR equation. Optional Show GN matrix toggles the kernel heatmap.
  8. Output (Stage 5): Codeword d ribbon.
Try this: start with K=40, E=108 for N=128 and a readable butterfly. The pipeline is: Information → Interleaved (c′) → U-vector (Frozen/Info) → Polar Transform → Codeword (d).

Parameters

  • K: Number of information bits (payload). Max 164 per 3GPP interleaver table.
  • E: Rate matching length (codeword length after puncturing/repetition). Used with K to select mother code size N.
  • N (mother code size): Computed as 2n (5 ≤ n ≤ 9) from 3GPP rules. Length of U-vector and codeword before rate matching.
  • Generate Random Bits & Run: Fills K random bits, computes N, interleaves, maps to sub-channels, applies polar transform, and updates all visualizations.

Limitations

  • Encoding only. This page builds the codeword; it does not implement the decoder (Successive Cancellation / SC-List), which is where polar codes' performance and complexity actually live.
  • No rate matching or CRC. The visualization stops at the mother codeword d of length N; sub-block interleaving, bit selection (puncturing/shortening/repetition) to length E, and CRC/PC distributed bits are described but not fully animated (see [[polarcodingratematch]]-style stages elsewhere).
  • Uplink (PUCCH/PUSCH-style) limits. K is capped at 164 per the interleaver table shown; the downlink (PBCH/PDCCH) construction and its different parameters are not covered.
  • Fixed reliability sequence. The frozen-set choice uses the single 3GPP master reliability table Q0Nmax; channel-specific or density-evolution-optimized frozen sets are not explored.
  • Noiseless, bit-exact. There is no channel, LLRs, or error injection — it shows the deterministic algebra of the transform, not coded performance (BLER vs SNR).
  • Display-bounded N. Large N produces a butterfly too dense to read; the tool is best used at small N (e.g. 128) for pedagogy.