Best Known (162−13, 162, s)-Nets in Base 3
(162−13, 162, 1594330)-Net over F3 — Constructive and digital
Digital (149, 162, 1594330)-net over F3, using
- net defined by OOA [i] based on linear OOA(3162, 1594330, F3, 14, 13) (dual of [(1594330, 14), 22320458, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3162, 4782991, F3, 2, 13) (dual of [(4782991, 2), 9565820, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3162, 4782992, F3, 2, 13) (dual of [(4782992, 2), 9565822, 14]-NRT-code), using
- trace code [i] based on linear OOA(981, 2391496, F9, 2, 13) (dual of [(2391496, 2), 4782911, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(981, 4782992, F9, 13) (dual of [4782992, 4782911, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(981, 4782993, F9, 13) (dual of [4782993, 4782912, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(978, 4782969, F9, 13) (dual of [4782969, 4782891, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(957, 4782969, F9, 10) (dual of [4782969, 4782912, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(93, 24, F9, 2) (dual of [24, 21, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(981, 4782993, F9, 13) (dual of [4782993, 4782912, 14]-code), using
- OOA 2-folding [i] based on linear OA(981, 4782992, F9, 13) (dual of [4782992, 4782911, 14]-code), using
- trace code [i] based on linear OOA(981, 2391496, F9, 2, 13) (dual of [(2391496, 2), 4782911, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3162, 4782992, F3, 2, 13) (dual of [(4782992, 2), 9565822, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(3162, 4782991, F3, 2, 13) (dual of [(4782991, 2), 9565820, 14]-NRT-code), using
(162−13, 162, large)-Net over F3 — Digital
Digital (149, 162, large)-net over F3, using
- 310 times duplication [i] based on digital (139, 152, large)-net over F3, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3152, large, F3, 13) (dual of [large, large−152, 14]-code), using
- 31 times code embedding in larger space [i] based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 330−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- 31 times code embedding in larger space [i] based on linear OA(3121, large, F3, 13) (dual of [large, large−121, 14]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3152, large, F3, 13) (dual of [large, large−152, 14]-code), using
(162−13, 162, large)-Net in Base 3 — Upper bound on s
There is no (149, 162, large)-net in base 3, because
- 11 times m-reduction [i] would yield (149, 151, large)-net in base 3, but