Best Known (81, 81+18, s)-Nets in Base 3
(81, 81+18, 731)-Net over F3 — Constructive and digital
Digital (81, 99, 731)-net over F3, using
- net defined by OOA [i] based on linear OOA(399, 731, F3, 18, 18) (dual of [(731, 18), 13059, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(399, 6579, F3, 18) (dual of [6579, 6480, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(398, 6578, F3, 18) (dual of [6578, 6480, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(397, 6561, F3, 19) (dual of [6561, 6464, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(31, 17, F3, 1) (dual of [17, 16, 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 X applied to Ce(18) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(398, 6578, F3, 18) (dual of [6578, 6480, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(399, 6579, F3, 18) (dual of [6579, 6480, 19]-code), using
(81, 81+18, 3289)-Net over F3 — Digital
Digital (81, 99, 3289)-net over F3, using
- 31 times duplication [i] based on digital (80, 98, 3289)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(398, 3289, F3, 2, 18) (dual of [(3289, 2), 6480, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(398, 6578, F3, 18) (dual of [6578, 6480, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(397, 6561, F3, 19) (dual of [6561, 6464, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(31, 17, F3, 1) (dual of [17, 16, 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 X applied to Ce(18) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(398, 6578, F3, 18) (dual of [6578, 6480, 19]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(398, 3289, F3, 2, 18) (dual of [(3289, 2), 6480, 19]-NRT-code), using
(81, 81+18, 367320)-Net in Base 3 — Upper bound on s
There is no (81, 99, 367321)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 171794 476419 749490 320580 150794 825684 402904 524051 > 399 [i]