The Vault

Application of puncturing of a CRC code for WLANs
Research Paper / Feb 2014

 

Application of puncturing of a CRC code for WLANs  

Monisha Ghosh, Senior Member, IEEE and Frank LaSita, Member, IEEE 

 

{monisha.ghosh, frank.lasita}@interdigital.com 

InterDigital Communications, LLC 

 

 

 

 

Abstract— In some cases, like the proposed IEEE 802.11ah 

draft specification [3], there arises a need to have a short 

Cyclic Redundancy Check (CRC) code to detect errors on the 

information bits in the various signal fields. While an optimum 

CRC of the required length can be chosen for this, in order to 

minimize complexity it may be desirable to “puncture” an 

existing CRC generator to reduce the number of parity bits 

transmitted. In this letter we evaluate puncturing of a CRC-8 

generator to generate a 4-bit CRC such that the minimum 

Hamming distance of the resultant code is 2.  

 

 

Index Terms—CRC, puncturing, 802.11  

 

I. INTRODUCTION 

YCLIC Redundancy Codes (CRCs) are widely used in 

many wired and wireless standard specifications as a 

 

means of error detection. The simplest CRC is a 1-bit even-

parity as used for example in the IEEE 802.11a specification 

[1] to protect the information bits in the signal field (SIG 

bits). The CRC enables the receiver to detect whether the 

SIG bits were received error-free, or not. As such, it is only 

used for error detection, and not error correction. The 1-bit 

parity check ensures that all single bit errors will be 

detected. Since the SIG bits carry information such as the 

Modulation Coding Scheme (MCS) used and number of 

bytes in the following data packet, it is very important that 

these bits be decoded correctly. The SIG field is usually 

transmitted at the lowest supported MCS for maximum 

robustness. 

 

CRCs of various lengths have been extensively studied 

and computer searches have been performed to determine 

optimal generator polynomials. However, the minimum 

Hamming distance of a CRC depends not only on the 

generator polynomial but also on the length of the 

information bits being protected by the CRC. It has been 

shown in [2] that a number of commonly used CRC 

generators are in fact suboptimal from a minimum 

Hamming distance point of view for certain data lengths. 

 

Recently, IEEE 802.11ah [3] has begun developing a 

physical (PHY) and medium access control (MAC) layer 

specification for the Sub-1 GHz non television white spaces 

(non-TVWS) frequencies in the 902-928 MHz [3]. The PHY 

is proposed to be a down clocked version of the IEEE 

802.11ac PHY which is the standard for the 5 GHz ISM 

band [4]. 802.11ac specifies an 8-bit CRC-8 for the SIG 

field with generator polynomial 𝑥! + 𝑥! + 𝑥 + 1. However, 

the SIG field in 802.11ah [3] is redefined to include only a 

 

4-bit CRC. In order to keep the implementation complexity 

low, it was proposed in [5] to just take the 4 least significant 

bits (LSBs) of the 8-bit parity generated by the CRC-8 

polynomial. We show in this paper that this choice leads to a 

CRC with minimum Hamming distance of 1, i.e. there are 

certain single bit error patterns that cannot be detected. We 

also perform a codeword weight analysis and demonstrate 

that there are other puncturing choices that can deliver a 4-

bit CRC with Hamming distance 2 and hence are more 

suitable than the 4 LSBs. 

 

The paper is organized as follows: Section II provides a 

brief background of the IEEE 802.11ah specification. 

Section III develops the codeword weight analysis and 

Section IV presents analytical and simulation results using 

the puncture patterns in Section III. Finally, Section V 

concludes the paper. 

 

II. THE IEEE 802.11AH SPECIFICATION 

In order to accommodate different spectral allocations 

 

worldwide, IEEE 802.11ah specifies two mandatory modes 

of operation: a 1 MHz mode and a 2 MHz mode in the 900 

MHz frequency band. One of the primary functional 

requirements of the 802.11ah specification is that it shall 

support a range of 1 km with at least 100 kbps throughput. 

This capability will enable applications like remote 

monitoring and smart metering. In order to obtain this range, 

the specification proposes a new MCS called “MCS0-Rep2” 

which is a mode with rate ½ convolutional coding using the 

standard [133,171] convolutional code, with BPSK 

modulation, followed by a repetition of the coded symbol. 

This mode is specified only for the 1 MHz mode of 

operation. The lowest data rate for the 2 MHz mode is the 

“MCS0” mode which is simply rate ½ BPSK. Since the SIG 

field is transmitted with the lowest supported data rate, the 1 

MHz mode SIG field uses MCS0-Rep2, while the 2 MHz 

mode SIG field uses MCS0.  

 

A. 1 MHz SIG Field  

The 1 MHz mode uses a 32-point FFT with a carrier 

 

spacing of 31.25 kHz. Of the 32 available frequency 

carriers, 24 are used for data, 2 are in-band pilots, one is the 

DC which is not used and the rest are band-edge carriers. 

The SIG field is defined to have 26 bits of information, 

followed by a 4-bit CRC and 6 tail bits (to terminate the 

convolution code) for a total of 36 bits. Using MCS0-Rep2, 

the SIG field thus requires 6 OFDM symbols. In [5] it was 

proposed that the 4-bit CRC be derived from the CRC-8 

 

 

 

 

polynomial by using only the 4 LSBs. 

 

B. 2 MHz SIG Field 

The 2 MHz mode uses a 64-point FFT with a carrier 

 

spacing of 31.25 kHz. Of the 64 available frequency 

carriers, 52 are used for data, 4 are in-band pilots, one is the 

DC which is not used and the rest are band-edge carriers. 

Depending on the transmission mode (multi-user or single 

user), there are two SIG fields specified: SIGA and SIGB. 

The SIGA field is defined to have 38 bits of information, 

followed by a 4-bit CRC and 6 tail bits (to terminate the 

convolution code) for a total of 48 bits. Using MCS0, the 

SIGA field thus requires 2 OFDM symbols. In [5] it was 

proposed that the 4-bit CRC in SIGA be derived from the 8-

bit CRC generated by the CRC-8 polynomial by using only 

the 4 LSBs. The SIGB field carries less information bits and 

hence has room for the full 8-bit CRC derived from CRC-8. 

 

III. CODEWORD WEIGHT ANALYSIS 

Let us assume that there are 𝑁 information bits. Then, we 

 

can form the generator matrix, G, for the polynomial  𝑥! + 𝑥! + 𝑥 + 1 as 𝐺 =    𝐼! 𝑃  where 𝐼!  is an 𝑁  ×𝑁 

identity matrix and 𝑃  is a 𝑁  ×8  parity matrix. The 

corresponding parity check matrix, H, can then be written as 𝐻 =    𝑃! 𝐼! , where 𝐼! is an 8  ×8 identity matrix.  

 

In order to reduce the number of parity bits from 8 to 4, 4 

rows are removed from matrix 𝑃! to get a 4  ×𝑁 matrix (or 4 

columns may be removed from 𝑃 ) to obtain a matrix 

denoted as 𝑃!!. The new parity check matrix describing this 

punctured code is then given by 𝐻! =    𝑃!! 𝐼! , where 𝐼! is 

an 4  ×4 identity matrix. 

 

In order to avoid a minimum Hamming distance of 1 for a 

code created as described above, a requirement is that no 

column of the parity check matrix be composed of all zeros 

[6]. Therefore, when 4 rows are removed in the example 

above, it is desirable for the matrix 𝐻! not to have an all-

zero column. The complete Hamming Weight (HW) 

distribution of a code may be derived from the parity check 

matrix as follows: the number of codewords with HW = i is 

the number of combinations of i columns of the parity check 

matrix such that a linear combination is equal to a zero 

vector [6].  

 

For the CRC-8 code, there are 70 combinations of 4-bit 

puncturing patterns, i.e., there are 70 ways to puncture 8 

parity bits to obtain 4 parity bits. Let 𝑐!𝑐!𝑐!𝑐!𝑐!𝑐!𝑐!𝑐!  denote the 8 parity bits generated from the 

CRC-8 polynomial where 𝑐!  is the least significant bit 

(LSB) and 𝑐!  is the most significant bit (MSB). Table 1 

shows the HW distribution for 𝑁 = 26 information bits 

when a 4-bit CRC is derived by puncturing the CRC-8 code.  

Only those combinations that have minimum HW = 2 are 

shown, other than the MSBs and LSBs which have 

minimum HW of 1. As comparison, the HW distribution of 

the optimal 4-bit CRC generator using the CRC-4 

polynomial 𝑥! + 𝑥 + 1

  is also shown. Table 2, on the other 

hand, shows a HW distribution for 𝑁 = 38 information bits 

 

when a 4-bit CRC is derived by puncturing the CRC-8 code.   

 

 

CRC bits HW = 

 

 

HW = 

 

 

HW = 

 

 

HW = 

 

 

 HW = 

 

 

c7c6c5c4 

(LSBs) 

 

3  28  254  1663 8839  

 

c3c2c1c0 

(MSBs) 

 

2  24  247  1687  8969 

 

c3c2c1c0 

(optimal 

 

4-bit 

 

CRC) 

 

0 15 280 1785 8736 

 

c5c2c1c0  0  31  247  1658  9029  

 

c5c3c2c0  0  34  247  1635  9029  

 

c5c4c1c0  0  27  265  1670  8879  

 

c5c4c2c0  0  31  252  1657  8976  

 

c5c4c3c1  0  29  264  1658  8890  𝑁 c6c3c1c0  0  35  246  1648  8992  

c6c3c2c1  0  29  260  1662  8930  

 

c6c4c3c1  0  31  263  1646  8901  

 

c6c5c2c0  0  33  259  1640  8909  

 

c6c5c3c2  0  32  259  1645  8925  

 

c7c4c2c1  0  33  248  1646  9018  

 

c7c4c3c0  0  29  264  1658  8890  

 

c7c4c3c1  0  29  260  1662  8930  

 

c7c5c2c0  0  34  255  1635  8949  

 

c7c5c4c1  0  32  266  1629  8874  

 

c7c6c3c2  0  29  262  1674  8896  

 

Table 1: HW distribution when a 4-bit CRC is derived from 

the CRC-8 code for 26 information bits. 

 

 

 

 

 

 

 

CRC 

 

Bits  

 

HW = 

 

1  

 

HW = 

 

 

HW = 

 

 

HW = 

 

 

HW = 5 

 

c7c6c5c4 

(LSBs) 

 

3  52  707  6932  53267  

 

c3c2c1c0 

(MSBs) 

 

3  51  697  6947  53377  

 

 

 

c3c2c1c0 

(optimal 

 

4-bit 

 

CRC) 

 

0 39 765 7101 52761 

 

c5c3c2c0  0  65  692  6867  53564  

 

c5c4c2c0  0  58  711  6942  53325  

 

c6c4c3c1  0  56  720  6938  53292  

 

c6c5c2c0  0  62  704  6894  53426  

 

c6c5c3c2  0  58  717  6906  53339  

 

c7c4c2c1  0  65  699  6865  53501  

 

c7c5c4c1 0 60 721 6876 53323 

 

Table 2: HW distribution when a 4-bit CRC is derived from 

the CRC-8 code for 38 information bits. 

 

As shown in Tables 1 and 2, if the 4 CRC bits are chosen 

to be 𝑐!𝑐!𝑐!𝑐! (i.e., the four LSBs) or bits 𝑐!𝑐!𝑐!𝑐! (i.e., the 

four MSBs), then the minimum Hamming distance is 1, i.e. 

there are 2 to 3 single-bit error patterns that will not be 

detected by the resulting code. Hence these are not good 

choices for a CRC. Instead, there are other choices shown in 

Tables 1 and 2 that deliver a minimum Hamming distance of 

2 and hence are better choices for a CRC.  

 

We also observe that as the number of data bits, 𝑁 , 

increases, the number of combinations that result in a code 

with Hamming distance 2 decreases, e.g. from 16 to 7 when 𝑁  increases from 26 to 38. Figure 1 shows that if the 

number of information bits exceeds 56, then no puncturing 

combination will yield a 4-bit CRC with minimum 

Hamming distance 2 that is derived from CRC-8. Instead 

one would have to puncture a longer CRC. One such 

possibility is the CRC-32, with polynomial 𝑥!" + 𝑥!" +𝑥!" + 𝑥!! + 𝑥!" + 𝑥!" + 𝑥!! + 𝑥!" + 𝑥! + 𝑥! + 𝑥! +𝑥! + 𝑥! + 𝑥 + 1 that is used in 802.11ah as the frame-check 

sequence. Figure 1 shows that it is possible to obtain a 4-bit 

CRC by puncturing the CRC-32 code for a larger number of 

information bits, up to 148. 

 

Figure 1 Generating a 4-bit CRC by puncturing CRC-8 and 

CRC-32 

 

IV. RESULTS 

The probability of false positives (which is the same as 

 

the probability of undetected error) is the probability that the 

CRC indicates that the SIG bits have been received correctly 

when they have not, and should be as low as possible. A 

false positive detection on the CRC leads to unsuccessful 

attempts at trying to decode the data packet that follows the 

SIG, with erroneous information for example on MCS and 

length from the incorrectly decoded SIG bits. This leads to 

wasted power consumption. 

 

It is possible to use the closed-form expression for 

probability of undetected error, 𝑃!", in (1) below for the 

additive white Gaussian noise (AWGN) case where it is 

assumed that the information bits are uncoded, 𝑃!  is the 

probability of bit error in AWGN as a function of signal-to-

noise-ratio (SNR) given by 𝑃! = 0.5erfc 0.5SNR , erfc is 

the complementary error function, HW(𝑖) is the Hamming 

weight distribution of the code and 𝑁!   is the total number of 

coded bits (information + parity) [6]. 

 

 𝑃!" = HW(𝑖)𝑃!! 1 − 𝑃! !!!!!!!!                     (1) 

 

With 𝑁! = 42, we evaluate equation (1) above and plot 

 

the results in Figure 2 for the optimal CRC-4 code, the 

CRC-8 code with the 4 LSB puncture pattern 𝑐!𝑐!𝑐!𝑐! and 

the CRC-8 code with puncture pattern 𝑐!𝑐!𝑐!𝑐!. It is clear 

that the 4 LSB puncture pattern performs significantly worse 

than the pattern 𝑐!𝑐!𝑐!𝑐! which is very close in performance 

to the optimal CRC-4. However, it may be argued that bit 

interleaved coded modulation (BICM) is used on the SIG 

bits in 802.11ah and hence the result for uncoded bits in 

Figure 2 does not apply there. Hence, in order to evaluate 

the various puncture patterns obtained from Section III 

using BICM, simulations were performed to obtain the 

probability of false positives. 

 

 

 

 

Figure 2 Theoretical 𝑃𝑢𝑑 performance. 

 

From Tables 1 and 2 above, we see that the 7 

combinations of parity bits in Table 2 are a subset of those 

in Table 1. Since it would be desirable to have the same 

puncture pattern for both 1 MHz and 2 MHz modes, these 7, 

along with the 4 MSBs, 4 LSBS and the optimum CRC-4 

were simulated in order to choose a common puncture 

pattern. 

 

Figure 3 shows the false positive rate for the 1 MHz mode 

in AWGN. The SIG field was generated according to the 

description in Section II, and ideal channel estimation and 

detection were assumed. The SIG error rate does not depend 

on the CRC chosen, but we see that there is almost a 1 dB 

difference in the false positive rate performance between the 

4 LSBs, as proposed in [5], and the best pattern which is 𝑐!𝑐!𝑐!𝑐! . This pattern also gives almost the same 

performance as the optimal CRC-4 generator. 

 

Figure 4 shows the results obtained with the 2 MHz mode 

in AWGN with the same 7 patterns, with the SIG field being 

generated according to the description in Section II. The 

spread in performance is less than in Figure 3, but 

nevertheless, the pattern 𝑐!𝑐!𝑐!𝑐! is almost as good as the 

optimum CRC-4 generator. 

 

 

 

 

Figure 3 False Positive Performance in the 1 MHz mode 

 

 

 

SIG Error Rate 

 

False Positive Rate 

 

 

 

 

Figure 4 False Positive Performance in the 2 MHz mode 

 

V. CONCLUSIONS 

We have shown analytically that puncturing the 8-bit 

 

CRC generated by the polynomial 𝑥! + 𝑥! + 𝑥 + 1  by 

keeping only the 4 LSBs of the 8 parity bits leads to a code 

with Hamming distance 1. This is a poor choice for any 

error detecting CRC and should not be used. Instead, we 

have shown, analytically and with simulations, that better 

puncturing choices are available that will result in a 

punctured code with Hamming distance 2. This allows the 

use of a shorter CRC without requiring a different CRC 

generator polynomial. We also show that for lengths beyond 

56 bits it is not possible to obtain a Hamming distance 2 

code by puncturing CRC-8 and instead puncturing the CRC-

32 could be considered. 

 

REFERENCES 

[1] Local and metropolitan area networks—Specific requirements; Part 

 

11: Wireless LAN Medium Access Control (MAC) and Physical 

Layer (PHY) specifications; High-speed Physical Layer in the 5 GHz 

Band, Available: http://easy.intranet.gr/IEEE80211a.pdf  

 

[2] P. Koopman and T. Chakravarty, “Cyclic Redundancy Code (CRC) 

polynomial selection for embedded networks,” Available: 

http://www.ece.cmu.edu/~koopman/roses/dsn04/koopman04_crc_pol

y_embedded.pdf . 

 

[3] “Proposed specification framework for Tgah,” 

https://mentor.ieee.org/802.11/dcn/11/11-11-1137-11-00ah-

specification-framework-for-tgah.docx . 

 

[4] Local and metropolitan area networks—Specific requirements; Part 

11: Wireless LAN Medium Access Control (MAC) and Physical 

Layer (PHY) specifications; Amendment 4: Enhancements for Very 

High Throughput for Operation in Bands below 6 GHz, IEEE 

P802.11ac/D3.1, August 2012. 

http://www.ieee802.org/11/private/Draft_Standards/11ac/DraftP802.1

1ac_D3.1.pdf. 

 

[5]  “SIG Field 4-bit CRC,” https://mentor.ieee.org/802.11/dcn/12/11-12-

0596-01-00ah-sig-field-4-bit-crc.pptx 

 

[6] S. Lin and D. J. Costello, Error Control Coding (2nd Edition), 

Prentice Hall, 2004. 

 

 

SIG Error Rate 

 

False Positive Rate