5.3.1 Polar coding

38.2123GPPMultiplexing and channel codingNRRelease 17TS

Tools: ARFCN - Frequency Conversion for 5G NR/LTE/UMTS/GSM

The bit sequence input for a given code block to channel coding is denoted by, where is the number of bits to encode. After encoding the bits are denoted by, where and the value of is determined by the following:

Denote by the rate matching output sequence length as given in Clause 5.4.1;

If and

;

else

;

end if

;

where .

UE is not expected to be configured with , where is the number of parity check bits defined in Clause 5.3.1.2.

5.3.1.1 Interleaving

The bit sequence is interleaved into bit sequence as follows:

where the interleaving pattern is given by the following:

else

;

for to

;

end if

end for

end if

where is given by Table 5.3.1.1-1 and .

Table 5.3.1.1-1: Interleaving pattern


0	0	28	67	56	122	84	68	112	33	140	38
1	2	29	69	57	123	85	73	113	36	141	144
2	4	30	70	58	126	86	78	114	44	142	39
3	7	31	71	59	127	87	84	115	47	143	145
4	9	32	72	60	129	88	90	116	64	144	40
5	14	33	76	61	132	89	92	117	74	145	146
6	19	34	77	62	134	90	94	118	79	146	41
7	20	35	81	63	138	91	96	119	85	147	147
8	24	36	82	64	139	92	99	120	97	148	148
9	25	37	83	65	140	93	102	121	100	149	149
10	26	38	87	66	1	94	105	122	103	150	150
11	28	39	88	67	3	95	107	123	117	151	151
12	31	40	89	68	5	96	109	124	125	152	152
13	34	41	91	69	8	97	112	125	131	153	153
14	42	42	93	70	10	98	114	126	136	154	154
15	45	43	95	71	15	99	116	127	142	155	155
16	49	44	98	72	21	100	121	128	12	156	156
17	50	45	101	73	27	101	124	129	17	157	157
18	51	46	104	74	29	102	128	130	23	158	158
19	53	47	106	75	32	103	130	131	37	159	159
20	54	48	108	76	35	104	133	132	48	160	160
21	56	49	110	77	43	105	135	133	75	161	161
22	58	50	111	78	46	106	141	134	80	162	162
23	59	51	113	79	52	107	6	135	86	163	163
24	61	52	115	80	55	108	11	136	137
25	62	53	118	81	57	109	16	137	143
26	65	54	119	82	60	110	22	138	13
27	66	55	120	83	63	111	30	139	18

5.3.1.2 Polar encoding

The Polar sequence is given by Table 5.3.1.2-1, where denotes a bit index before Polar encoding for and . The Polar sequence is in ascending order of reliability , where denotes the reliability of bit index .

For any code block encoded to bits, a same Polar sequence is used. The Polar sequence is a subset of Polar sequence with all elements of values less than , ordered in ascending order of reliability .

Denote as a set of bit indices in Polar sequence , and as the set of other bit indices in Polar sequence , where and are given in Clause 5.4.1.1, , , and is the number of parity check bits.

Denote as the -th Kronecker power of matrix , where .

For a bit index with , denote as the -th row of and as the row weight of , where is the number of ones in . Denote the set of bit indices for parity check bits as , where . A number of parity check bits are placed in the least reliable bit indices in . A number of other parity check bits are placed in the bit indices of minimum row weight in , where denotes the most reliable bit indices in ; if there are more than bit indices of the same minimum row weight in , the other parity check bits are placed in the bit indices of the highest reliability and the minimum row weight in .

Generate according to the following:

;

; ; ; ; ;

for to

; ; ; ; ; ;

;

else

;

end if

else

;

end if

end for

else

for to

;

else

;

end if

end for

end if

The output after encoding is obtained by . The encoding is performed in GF(2).

Table 5.3.1.2-1: Polar sequence and its corresponding reliability


0	0	128	518	256	94	384	214	512	364	640	414	768	819	896	966
1	1	129	54	257	204	385	309	513	654	641	223	769	814	897	755
2	2	130	83	258	298	386	188	514	659	642	663	770	439	898	859
3	4	131	57	259	400	387	449	515	335	643	692	771	929	899	940
4	8	132	521	260	608	388	217	516	480	644	835	772	490	900	830
5	16	133	112	261	352	389	408	517	315	645	619	773	623	901	911
6	32	134	135	262	325	390	609	518	221	646	472	774	671	902	871
7	3	135	78	263	533	391	596	519	370	647	455	775	739	903	639
8	5	136	289	264	155	392	551	520	613	648	796	776	916	904	888
9	64	137	194	265	210	393	650	521	422	649	809	777	463	905	479
10	9	138	85	266	305	394	229	522	425	650	714	778	843	906	946
11	6	139	276	267	547	395	159	523	451	651	721	779	381	907	750
12	17	140	522	268	300	396	420	524	614	652	837	780	497	908	969
13	10	141	58	269	109	397	310	525	543	653	716	781	930	909	508
14	18	142	168	270	184	398	541	526	235	654	864	782	821	910	861
15	128	143	139	271	534	399	773	527	412	655	810	783	726	911	757
16	12	144	99	272	537	400	610	528	343	656	606	784	961	912	970
17	33	145	86	273	115	401	657	529	372	657	912	785	872	913	919
18	65	146	60	274	167	402	333	530	775	658	722	786	492	914	875
19	20	147	280	275	225	403	119	531	317	659	696	787	631	915	862
20	256	148	89	276	326	404	600	532	222	660	377	788	729	916	758
21	34	149	290	277	306	405	339	533	426	661	435	789	700	917	948
22	24	150	529	278	772	406	218	534	453	662	817	790	443	918	977
23	36	151	524	279	157	407	368	535	237	663	319	791	741	919	923
24	7	152	196	280	656	408	652	536	559	664	621	792	845	920	972
25	129	153	141	281	329	409	230	537	833	665	812	793	920	921	761
26	66	154	101	282	110	410	391	538	804	666	484	794	382	922	877
27	512	155	147	283	117	411	313	539	712	667	430	795	822	923	952
28	11	156	176	284	212	412	450	540	834	668	838	796	851	924	495
29	40	157	142	285	171	413	542	541	661	669	667	797	730	925	703
30	68	158	530	286	776	414	334	542	808	670	488	798	498	926	935
31	130	159	321	287	330	415	233	543	779	671	239	799	880	927	978
32	19	160	31	288	226	416	555	544	617	672	378	800	742	928	883
33	13	161	200	289	549	417	774	545	604	673	459	801	445	929	762
34	48	162	90	290	538	418	175	546	433	674	622	802	471	930	503
35	14	163	545	291	387	419	123	547	720	675	627	803	635	931	925
36	72	164	292	292	308	420	658	548	816	676	437	804	932	932	878
37	257	165	322	293	216	421	612	549	836	677	380	805	687	933	735
38	21	166	532	294	416	422	341	550	347	678	818	806	903	934	993
39	132	167	263	295	271	423	777	551	897	679	461	807	825	935	885
40	35	168	149	296	279	424	220	552	243	680	496	808	500	936	939
41	258	169	102	297	158	425	314	553	662	681	669	809	846	937	994
42	26	170	105	298	337	426	424	554	454	682	679	810	745	938	980
43	513	171	304	299	550	427	395	555	318	683	724	811	826	939	926
44	80	172	296	300	672	428	673	556	675	684	841	812	732	940	764
45	37	173	163	301	118	429	583	557	618	685	629	813	446	941	941
46	25	174	92	302	332	430	355	558	898	686	351	814	962	942	967
47	22	175	47	303	579	431	287	559	781	687	467	815	936	943	886
48	136	176	267	304	540	432	183	560	376	688	438	816	475	944	831
49	260	177	385	305	389	433	234	561	428	689	737	817	853	945	947
50	264	178	546	306	173	434	125	562	665	690	251	818	867	946	507
51	38	179	324	307	121	435	557	563	736	691	462	819	637	947	889
52	514	180	208	308	553	436	660	564	567	692	442	820	907	948	984
53	96	181	386	309	199	437	616	565	840	693	441	821	487	949	751
54	67	182	150	310	784	438	342	566	625	694	469	822	695	950	942
55	41	183	153	311	179	439	316	567	238	695	247	823	746	951	996
56	144	184	165	312	228	440	241	568	359	696	683	824	828	952	971
57	28	185	106	313	338	441	778	569	457	697	842	825	753	953	890
58	69	186	55	314	312	442	563	570	399	698	738	826	854	954	509
59	42	187	328	315	704	443	345	571	787	699	899	827	857	955	949
60	516	188	536	316	390	444	452	572	591	700	670	828	504	956	973
61	49	189	577	317	174	445	397	573	678	701	783	829	799	957	1000
62	74	190	548	318	554	446	403	574	434	702	849	830	255	958	892
63	272	191	113	319	581	447	207	575	677	703	820	831	964	959	950
64	160	192	154	320	393	448	674	576	349	704	728	832	909	960	863
65	520	193	79	321	283	449	558	577	245	705	928	833	719	961	759
66	288	194	269	322	122	450	785	578	458	706	791	834	477	962	1008
67	528	195	108	323	448	451	432	579	666	707	367	835	915	963	510
68	192	196	578	324	353	452	357	580	620	708	901	836	638	964	979
69	544	197	224	325	561	453	187	581	363	709	630	837	748	965	953
70	70	198	166	326	203	454	236	582	127	710	685	838	944	966	763
71	44	199	519	327	63	455	664	583	191	711	844	839	869	967	974
72	131	200	552	328	340	456	624	584	782	712	633	840	491	968	954
73	81	201	195	329	394	457	587	585	407	713	711	841	699	969	879
74	50	202	270	330	527	458	780	586	436	714	253	842	754	970	981
75	73	203	641	331	582	459	705	587	626	715	691	843	858	971	982
76	15	204	523	332	556	460	126	588	571	716	824	844	478	972	927
77	320	205	275	333	181	461	242	589	465	717	902	845	968	973	995
78	133	206	580	334	295	462	565	590	681	718	686	846	383	974	765
79	52	207	291	335	285	463	398	591	246	719	740	847	910	975	956
80	23	208	59	336	232	464	346	592	707	720	850	848	815	976	887
81	134	209	169	337	124	465	456	593	350	721	375	849	976	977	985
82	384	210	560	338	205	466	358	594	599	722	444	850	870	978	997
83	76	211	114	339	182	467	405	595	668	723	470	851	917	979	986
84	137	212	277	340	643	468	303	596	790	724	483	852	727	980	943
85	82	213	156	341	562	469	569	597	460	725	415	853	493	981	891
86	56	214	87	342	286	470	244	598	249	726	485	854	873	982	998
87	27	215	197	343	585	471	595	599	682	727	905	855	701	983	766
88	97	216	116	344	299	472	189	600	573	728	795	856	931	984	511
89	39	217	170	345	354	473	566	601	411	729	473	857	756	985	988
90	259	218	61	346	211	474	676	602	803	730	634	858	860	986	1001
91	84	219	531	347	401	475	361	603	789	731	744	859	499	987	951
92	138	220	525	348	185	476	706	604	709	732	852	860	731	988	1002
93	145	221	642	349	396	477	589	605	365	733	960	861	823	989	893
94	261	222	281	350	344	478	215	606	440	734	865	862	922	990	975
95	29	223	278	351	586	479	786	607	628	735	693	863	874	991	894
96	43	224	526	352	645	480	647	608	689	736	797	864	918	992	1009
97	98	225	177	353	593	481	348	609	374	737	906	865	502	993	955
98	515	226	293	354	535	482	419	610	423	738	715	866	933	994	1004
99	88	227	388	355	240	483	406	611	466	739	807	867	743	995	1010
100	140	228	91	356	206	484	464	612	793	740	474	868	760	996	957
101	30	229	584	357	95	485	680	613	250	741	636	869	881	997	983
102	146	230	769	358	327	486	801	614	371	742	694	870	494	998	958
103	71	231	198	359	564	487	362	615	481	743	254	871	702	999	987
104	262	232	172	360	800	488	590	616	574	744	717	872	921	1000	1012
105	265	233	120	361	402	489	409	617	413	745	575	873	501	1001	999
106	161	234	201	362	356	490	570	618	603	746	913	874	876	1002	1016
107	576	235	336	363	307	491	788	619	366	747	798	875	847	1003	767
108	45	236	62	364	301	492	597	620	468	748	811	876	992	1004	989
109	100	237	282	365	417	493	572	621	655	749	379	877	447	1005	1003
110	640	238	143	366	213	494	219	622	900	750	697	878	733	1006	990
111	51	239	103	367	568	495	311	623	805	751	431	879	827	1007	1005
112	148	240	178	368	832	496	708	624	615	752	607	880	934	1008	959
113	46	241	294	369	588	497	598	625	684	753	489	881	882	1009	1011
114	75	242	93	370	186	498	601	626	710	754	866	882	937	1010	1013
115	266	243	644	371	646	499	651	627	429	755	723	883	963	1011	895
116	273	244	202	372	404	500	421	628	794	756	486	884	747	1012	1006
117	517	245	592	373	227	501	792	629	252	757	908	885	505	1013	1014
118	104	246	323	374	896	502	802	630	373	758	718	886	855	1014	1017
119	162	247	392	375	594	503	611	631	605	759	813	887	924	1015	1018
120	53	248	297	376	418	504	602	632	848	760	476	888	734	1016	991
121	193	249	770	377	302	505	410	633	690	761	856	889	829	1017	1020
122	152	250	107	378	649	506	231	634	713	762	839	890	965	1018	1007
123	77	251	180	379	771	507	688	635	632	763	725	891	938	1019	1015
124	164	252	151	380	360	508	653	636	482	764	698	892	884	1020	1019
125	768	253	209	381	539	509	248	637	806	765	914	893	506	1021	1021
126	268	254	284	382	111	510	369	638	427	766	752	894	749	1022	1022
127	274	255	648	383	331	511	190	639	904	767	868	895	945	1023	1023