Best Known (149−18, 149, s)-Nets in Base 3
(149−18, 149, 59052)-Net over F3 — Constructive and digital
Digital (131, 149, 59052)-net over F3, using
- t-expansion [i] based on digital (130, 149, 59052)-net over F3, using
- net defined by OOA [i] based on linear OOA(3149, 59052, F3, 19, 19) (dual of [(59052, 19), 1121839, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3149, 531469, F3, 19) (dual of [531469, 531320, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3145, 531441, F3, 19) (dual of [531441, 531296, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3121, 531441, F3, 16) (dual of [531441, 531320, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(34, 28, F3, 2) (dual of [28, 24, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(3149, 531469, F3, 19) (dual of [531469, 531320, 20]-code), using
- net defined by OOA [i] based on linear OOA(3149, 59052, F3, 19, 19) (dual of [(59052, 19), 1121839, 20]-NRT-code), using
(149−18, 149, 177156)-Net over F3 — Digital
Digital (131, 149, 177156)-net over F3, using
- 32 times duplication [i] based on digital (129, 147, 177156)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3147, 177156, F3, 3, 18) (dual of [(177156, 3), 531321, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3147, 531468, F3, 18) (dual of [531468, 531321, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3146, 531467, F3, 18) (dual of [531467, 531321, 19]-code), using
- construction X4 applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3145, 531441, F3, 19) (dual of [531441, 531296, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(3121, 531441, F3, 16) (dual of [531441, 531320, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(325, 26, F3, 25) (dual of [26, 1, 26]-code or 26-arc in PG(24,3)), using
- dual of repetition code with length 26 [i]
- linear OA(31, 26, F3, 1) (dual of [26, 25, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(18) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3146, 531467, F3, 18) (dual of [531467, 531321, 19]-code), using
- OOA 3-folding [i] based on linear OA(3147, 531468, F3, 18) (dual of [531468, 531321, 19]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3147, 177156, F3, 3, 18) (dual of [(177156, 3), 531321, 19]-NRT-code), using
(149−18, 149, large)-Net in Base 3 — Upper bound on s
There is no (131, 149, large)-net in base 3, because
- 16 times m-reduction [i] would yield (131, 133, large)-net in base 3, but