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Two-weight codes

Columns description:

(1) Parameters of the code
(2) Far from Griesmer bound (the optimals are denoted by *)
(3) Weight enumerator
(4) The number of nonequivalence codes
(5) The number of projective self-dual codes
(6) |Aut|
(7) Information about known examples of codes with such parameters, according to [1]
(8) More information about classification results

Table 1. Binary projective double-weight codes

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
[5,4,2]₂	0 ^*	1 + 10z² + 5z⁴	1	1	120
[6,4,2]₂	1	1 + 6z² + 9z⁴	1	1	72	SU2
[14,6,4]₂	4	1 + 14z⁴ + 49z⁸	1	1	56448	SU2
[21,6,8]₂	4	1 + 21z⁸ + 42z¹²	2	2	336,1008	SU2
[27,6,12]₂	2	1 + 36z¹² + 27z¹⁶	5	5	24,120,160,384,51840	RT2	[4]
[28,6,12]₂	3	1 + 28z¹² + 35z¹⁶	7	7	24,120,84,96,1344,384,40320	SU2	[4]
[30,8,18]₂	11	1 + 30z⁸ + 225z¹⁶	1	1	812851200	CY4
[45,8,16]₂	11	1 + 45z¹⁶ + 210z²⁴	2	2	120960,3628800	CY4
[51,8,24]₂	1 ^*	1 + 204z²⁴ + 51z³²	1	1	48960
[60,8,24]₂	10	1 + 60z²⁴ + 195z³²	12	12	192,16,24,32,40,96,14400,576,120,288,4320,720	CY4
[68,8,32]₂	3 ^*	1 + 187z³² + 68z⁴⁰	41	29	2⁷,6²,48³,12,192,96,16320,1¹⁵,3³,4⁴,8²,16

Table 2. Ternary projective double-weight codes

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
[10,4,6]₃	0 ^*	1 + 60z⁶ + 20z⁹	1	1	2880
[12,4,6]₃	2	1 + 24z⁶ + 56z⁹	2	2	288,1152	CY4
[15,4,9]₃	1 ^*	1 + 50z⁹ + 30z¹²	2	2	72,1440		[2]
[16,4,9]₃	2	1 + 32z⁹ + 48z¹²	4	4	144²,2304,64	CY4
[20,4,12]₃	1	1 + 40z¹² + 40z¹⁵	4	4	24,480,128,160	CY4
[11,5,6]₃	0 ^*	1 + 132z⁶ + 110z⁹	1	0	15840
[55,5,36]₃	0 ^*	1 + 220z³⁶ + 22z⁴⁵	1	0	15840		[3]
[56,6,36]₃	0 ^*	1 + 616z³⁶ + 112z⁴⁵	1	1	80640		[3]

Table 3. Projective double-weight codes over F₄

(1)	(2)	(3)	(4)	(5)	(6)	(7)
[6,3,4]₄	0 ^*	1 + 45z⁴ + 18z⁶	1	1	2160
[7,3,4]₄	1	1 + 21z⁴ + 42z⁶	1	1	1008	CY4
[9,3,6]₄	0 ^*	1 + 36z⁶ + 27z⁸	1	1	1296	RT3
[10,4,4]₄	3	1 + 30z⁴ + 225z⁸	1	1	259200	CY4
[15,4,8]₄	3	1 + 45z⁸ + 210z¹²	2	2	2160,120960	CY4
[17,4,12]₄	0 ^*	1 + 204z¹² + 51z¹⁶	1	1	48960
[20,4,12]₄	3	1 + 60z¹² + 195z¹⁶	7	7	1080,432,288,4320,576,720,120	CY4
[25,4,16]₄	3	1 + 75z¹⁶ + 180z²⁰	19	19	144,576,288,2160,48,72,360,43200,24²,18,12²,60,36³,120,96	CY4
[30,4,20]₄	2	1 + 90z²⁰ + 165z²⁴	68	66	18⁷,72²,24⁸,6²⁰,3⁴, 12¹¹,288,144²,48²,36², 180²,720,9
[34,4,24]₄	1 ^*	1 + 153z²⁴ + 102z²⁸	84	38	6²⁴,3³²,12⁷,18⁹,72³, 48,144,24³,36²,120,204
[35,4,24]₄	2	1 + 105z²⁴ + 150z²⁸	231	179	12²⁵,3⁷⁸,6⁶⁸,48⁴,72⁵, 36¹⁰,24⁷,21,288,720, 60³,18⁹,216,1200,96, 144²,9⁸,108,30,63,90²,42	CY4
[40,4,28]₄	2	1 +120z²⁸ + 135z³²	481	315	3²³⁹,6¹²⁹,9⁷,12⁴²,48⁶, 18¹²,36¹⁰,24¹⁶,72²,60, 432,5760,1152,576²,180², 30²,144²,96,120²,480, 384,155520	CY4

Table 4. Projective double-weight codes over F₅

(1)	(2)	(3)	(4)	(5)	(6)	(7)
[12,4,5]₅	4	1 +48z⁵ + 576z¹⁰	1	1	460800	CY4
[18,4,10]₅	4	1 +72z¹⁰ + 552z¹⁵	1	1	2880	CY4
[24,4,15]₅	4	1 +96z¹⁵ + 528z²⁰	7	7	160,192,128,576,96,3840,4608	CY4
[26,4,20]₅	0 ^*	1 +520z²⁰ + 104z²⁵	1	1	124800	RT2
[30,4,20]₅	4	1 +120z²⁰ + 504z²⁵	38	38	8⁸,4,32⁵,40,16⁴, 480,48⁶,96²,80,128, 288,400,384,9600,160,64,240,800	CY4
[39,4,30]₅	0 ^*	1 +468z³⁰ + 156z³⁵	8	8	4,16,24²,72,12,48²

REFERENCES

[1] A. R. Calderbank, W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc. 18 (1986) pp. 97-122.
[2] N.Hamada and T. Helleseth. A characterization of some {3ν₂+ν₃, 3ν₁ + ν₂; 3, 3}-minihypers and some [15,4,9;3]-codes with B₂=0. J. Stat. Plann. Infer., 56:129-146, 1996.
[3] R. Hill, Caps and codes, Discrete Math., 22 (1978) pp. 111-137.
[4] V.D. Tonchev, The uniformly packed binary [27,21,3] and [35,29,3] codes. Discrete Math., 149, (1996) pp. 283-288.