On the Measurement of the (Non)linearity of Costas Permutations

Table 2

Linearity results for 𝑊₁- (a) and 𝐺₂- (b) permutations generated in 𝔽(𝑝), 7≤𝑝≤151: the columns correspond from left to right to 𝑝, the minimal and maximal linearity observed, and the mean and standard deviations of the linearities.

(a)

𝑝	min 𝕃	max 𝕃	mean 𝕃	std 𝕃
7	3.8344	3.8344	3.8344	0
11	4.7015	6.5598	5.9362	0.6950
13	5.1913	7.5414	6.4948	0.8282
17	6.2187	9.3125	7.8076	0.7503
19	7.6955	11.4045	8.9887	0.8291
23	8.2347	13.9195	9.9797	1.1294
29	10.2502	16.8729	12.0515	1.4318
31	10.9677	16.0584	13.0976	1.4772
37	11.2892	20.5871	14.8351	2.1180
41	12.7289	21.6292	16.1072	2.2266
43	12.6109	26.9280	17.1595	2.7143
47	13.6820	28.3311	18.3678	2.5944
53	14.7483	29.0807	20.7323	3.4550
59	16.5240	33.8105	23.1238	3.7493
61	18.0203	35.1987	23.8661	3.5871
67	17.9924	38.6901	26.2275	4.0515
71	19.2378	36.4151	27.6938	3.9807
73	18.1823	43.1096	28.6351	4.7167
79	19.4471	41.3463	30.9327	4.2014
83	21.2388	45.4478	32.5626	4.2763
89	21.3421	50.3580	35.0311	4.6115
97	24.6606	53.2888	38.2512	4.6876
101	27.1879	52.6007	39.8379	5.0796
103	25.4178	55.2598	40.6539	5.5482
107	27.3222	57.9382	42.2601	5.4518
109	24.9826	59.9810	43.1139	5.5113
113	29.6132	60.6273	44.7029	5.6808
127	32.5151	70.1353	50.4963	5.3487
131	31.2879	70.7198	51.9830	5.7581
137	37.2955	71.0958	54.4337	5.7566
139	39.2845	71.4561	55.2887	6.3653
149	40.5434	76.0981	59.2918	6.0764
151	37.4170	81.0333	60.1070	6.0674

(b)

𝑝	min 𝕃	max 𝕃	mean 𝕃	std 𝕃
7	3.7880	4.8439	4.4465	0.4119
11	5.5053	7.4756	6.1462	0.4734
13	6.5154	8.5341	7.3052	0.6860
17	7.0501	11.0186	8.5779	0.8158
19	7.5172	12.9660	9.2807	1.1130
23	8.9254	15.2323	10.6936	1.2187
29	10.3331	18.3192	12.9565	1.8357
31	10.9192	19.9770	13.5635	1.6404
37	11.9643	24.3642	15.7800	2.5823
41	12.7863	25.1720	17.1771	2.8822
43	12.7419	25.3322	17.8591	3.0673
47	14.0212	28.1627	19.1228	3.2689
53	15.0066	33.3970	21.5939	3.8445
59	16.4316	35.5213	23.5966	4.3340
61	16.2902	36.1331	24.3684	4.5547
67	17.4479	39.8553	26.7267	4.9172
71	18.4225	40.7985	28.2850	5.0988
73	18.3244	42.1267	29.0482	5.1430
79	18.5635	47.8331	31.4486	5.5526
83	20.0101	49.1780	33.0352	5.6957
89	21.2534	51.4717	35.4218	5.5741
97	22.1986	57.7464	38.6617	6.2288
101	22.9944	56.8656	40.2868	6.6020
103	25.5362	57.2796	41.0855	6.5606
107	24.0941	63.6045	42.7078	6.5930
109	24.7888	64.2623	43.5123	6.6427
113	26.5017	64.3054	45.1397	7.0107
127	35.1980	65.9180	50.7644	6.1753
131	30.9777	73.9441	52.4322	7.4101
137	32.7740	77.6968	54.8584	7.5626
139	31.6170	79.5067	55.6664	7.5732
149	35.0306	83.2909	59.7193	7.8052
151	36.3423	84.3540	60.5394	8.0897