1. “Parity bits” can mean three different things

HARQ explanations often say “LDPC adds parity bits” and move on. That hides three distinct mechanisms:

  1. TB CRC bits — error detection at transport-block level;
  2. code-block CRC bits — error detection for segmented code blocks when applicable; and
  3. LDPC parity/redundancy bits — forward-error-correction redundancy used by the decoder.

Only the third category is the main source of the error-correcting redundancy exploited by incremental-redundancy HARQ.

2. Start from the transport block

MAC hands PHY a transport block:

Transport Block
      |
      v
TB CRC attachment
      |
      v
code-block segmentation if required
      |
      v
LDPC encoding

The TB CRC lets the receiver decide whether the recovered transport block is valid after decoding.

3. LDPC systematic and parity structure

Conceptually, an LDPC codeword contains information-related systematic bits and parity/redundancy bits:

+------------------------+---------------------------+
| systematic/data region | LDPC parity/redundancy   |
+------------------------+---------------------------+

The complete codeword satisfies the parity-check constraints

\[ Hc^T=0, \]

where \(H\) is the sparse LDPC parity-check matrix and \(c\) is the encoded codeword.

The parity bits are not a duplicate copy of the payload. They encode additional algebraic constraints that let the iterative decoder infer uncertain information bits.

4. Why not transmit the entire mother codeword every time?

The scheduled PDSCH/PUSCH resource allocation determines how many coded bits can actually be carried.

Suppose, purely as an example:

available encoded sequence = 12,000 bits
current allocation carries  =  6,000 bits

PHY therefore needs rate matching to select the required number of coded bits.

5. NR LDPC rate matching uses a circular-buffer model

After LDPC encoding and the standardized bit-ordering/interleaving steps, rate matching selects bits from a circular-buffer representation.

Conceptually:

                   circular buffer

+--------------------------------------------------+
| encoded positions: systematic + parity regions  |
+--------------------------------------------------+
           ^           ^           ^
         RV start    RV start    RV start

The actual NR layout includes punctured/null/filler handling and base-graph-dependent equations, so this diagram is deliberately conceptual.

The important principle is:

RV changes the starting position used for rate-matching bit selection.

6. The four redundancy versions

NR commonly uses:

RV = 0
RV = 1
RV = 2
RV = 3

TS 38.212 defines the base-graph-dependent starting position \(k_0\) for the selected RV. The selection then walks the circular buffer, skipping positions that must not be transmitted, until the required output length \(E\) is obtained.

Therefore RV is not a label such as:

RV0 = all data bits
RV1 = parity set A
RV2 = parity set B
RV3 = parity set C

That shortcut is too crude to be technically correct.

7. What incremental redundancy really means

Use a simplified encoded sequence:

D0 D1 D2 D3 D4 D5 P0 P1 P2 P3 P4 P5 P6 P7 ...

where D represents systematic/data-related positions and P represents parity positions.

A first transmission might expose a subset such as:

RV0 selection:
D0 D1 D2 D3 D4 D5 P0 P1 P2 ...

If decoding fails, a later transmission with another RV might expose a partly different subset:

RV2 selection:
D4 D5 P3 P4 P5 P6 P7 ...

The receiver can now gain both:

  • stronger evidence for overlapping positions; and
  • completely new parity/redundancy observations.

That is the essence of incremental redundancy.

Do not literalize the example: the exact selected positions depend on base graph, code-block parameters, circular-buffer size, allocated E bits, filler/null positions and the standardized RV start equations.

8. How parity information helps an uncertain data bit

Suppose the direct channel evidence for data bit \(D_3\) is almost neutral:

LLR(D3) = +0.05

A simplified parity constraint could look like

\[ D_1 \oplus D_3 \oplus D_7 \oplus P_4 = 0. \]

If the decoder has strong beliefs for \(D_1\), \(D_7\) and \(P_4\), that parity equation contributes information about \(D_3\).

Real LDPC decoding uses many sparse parity constraints simultaneously. Iterative message passing exchanges information between variable nodes and check nodes until the codeword converges or decoding stops.

variable-node beliefs
       <---->
parity-check constraints
       <---->
updated bit beliefs

A retransmission that supplies new parity observations can therefore make previously undecodable systematic bits recoverable.

9. Rate recovery maps each received LLR back to the right encoded position

Suppose transmission 1 gives:

encoded position C145 -> LLR +1.2
encoded position C146 -> LLR -0.7

and transmission 2 with another RV also contains C145:

C145 -> new LLR +2.0

Rate recovery knows that both observations correspond to the same encoded position and can combine them:

\[ L(C145)=1.2+2.0=3.2 \]

If transmission 2 contains a previously unseen position C903, the receiver gains its first observation for that parity/redundancy bit.

This is why different RV transmissions can still be combined coherently.

10. Chase combining versus incremental redundancy

Chase combining

The retransmission mostly repeats the same coded information:

TX1: A B C D
TX2: A B C D

Benefit: stronger soft evidence for the same positions.

Incremental redundancy

The retransmission includes at least partly different coded information:

TX1: A B C D
TX2: C D E F

Benefit: stronger evidence for C/D and new redundancy E/F.

NR’s RV/rate-matching framework supports the latter behavior naturally.

11. Does the transmitter have to store RV0/RV1/RV2/RV3 bitstreams?

Not architecturally.

At MAC level, the transmitter retains the same TB/MAC PDU in the HARQ process. PHY can regenerate the encoded representation and apply the requested RV on a retransmission.

A real modem implementation may cache encoded/intermediate data to save latency or energy, but that is an implementation optimization rather than a requirement that four complete coded copies be retained.

12. Full coding and HARQ chain

graph TD
    TB[MAC Transport Block] --> CRC[TB CRC]
    CRC --> CBS[Code-block segmentation]
    CBS --> LDPC[LDPC encoder]
    LDPC --> CB[Circular-buffer encoded representation]
    CB --> RV0[Rate matching / RV selection]
    RV0 --> MOD[Modulation and transmission]
    MOD --> RX[Receiver LLRs]
    RX --> RR[Rate recovery]
    RR --> SB[HARQ soft buffer]
    SB --> DEC[LDPC decode]
    DEC --> OK{CRC pass?}
    OK -- No --> RETX[Schedule another RV]
    RETX --> CB
    OK -- Yes --> DONE[TB delivered]

13. Why HARQ is called “hybrid”

Pure ARQ would merely detect failure and request another transmission.

HARQ combines:

FEC coding
+
soft information from failed reception
+
retransmission
+
optional new redundancy through another RV

The retransmission therefore does not discard the work already done by the decoder. It improves the same codeword’s evidence.

14. Standards trail

Primary references:

  • 3GPP TS 38.212 — DL-SCH/UL-SCH transport-block CRC, code-block segmentation, LDPC encoding, LDPC rate matching, circular-buffer bit selection and redundancy-version start positions.
  • 3GPP TS 38.214 — physical-layer procedures that use the RV and HARQ control information for PDSCH/PUSCH.

ETSI specification families:

The next article moves from one HARQ process to the whole pipeline: how many processes a UE can support, how many the network actually configures, what DCI can address, and why long NTN delays change the design.