Best Known (108−17, 108, s)-Nets in Base 3
(108−17, 108, 2465)-Net over F3 — Constructive and digital
Digital (91, 108, 2465)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (83, 100, 2461)-net over F3, using
- net defined by OOA [i] based on linear OOA(3100, 2461, F3, 17, 17) (dual of [(2461, 17), 41737, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3100, 19689, F3, 17) (dual of [19689, 19589, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3100, 19692, F3, 17) (dual of [19692, 19592, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- linear OA(3100, 19683, F3, 17) (dual of [19683, 19583, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(30, 9, F3, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3100, 19692, F3, 17) (dual of [19692, 19592, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3100, 19689, F3, 17) (dual of [19689, 19589, 18]-code), using
- net defined by OOA [i] based on linear OOA(3100, 2461, F3, 17, 17) (dual of [(2461, 17), 41737, 18]-NRT-code), using
- digital (0, 8, 4)-net over F3, using
(108−17, 108, 9859)-Net over F3 — Digital
Digital (91, 108, 9859)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3108, 9859, F3, 2, 17) (dual of [(9859, 2), 19610, 18]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3106, 9858, F3, 2, 17) (dual of [(9858, 2), 19610, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3106, 19716, F3, 17) (dual of [19716, 19610, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(3100, 19683, F3, 17) (dual of [19683, 19583, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(373, 19683, F3, 13) (dual of [19683, 19610, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(3106, 19716, F3, 17) (dual of [19716, 19610, 18]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3106, 9858, F3, 2, 17) (dual of [(9858, 2), 19610, 18]-NRT-code), using
(108−17, 108, 4530604)-Net in Base 3 — Upper bound on s
There is no (91, 108, 4530605)-net in base 3, because
- 1 times m-reduction [i] would yield (91, 107, 4530605)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1127 132033 787681 540550 394597 038183 977208 958899 195233 > 3107 [i]