Best Known (93, 114, s)-Nets in Base 3
(93, 114, 688)-Net over F3 — Constructive and digital
Digital (93, 114, 688)-net over F3, using
- 32 times duplication [i] based on digital (91, 112, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 28, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 28, 172)-net over F81, using
(93, 114, 3289)-Net over F3 — Digital
Digital (93, 114, 3289)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3114, 3289, F3, 2, 21) (dual of [(3289, 2), 6464, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3114, 6578, F3, 21) (dual of [6578, 6464, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3114, 6579, F3, 21) (dual of [6579, 6465, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(3113, 6562, F3, 21) (dual of [6562, 6449, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [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 C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3114, 6579, F3, 21) (dual of [6579, 6465, 22]-code), using
- OOA 2-folding [i] based on linear OA(3114, 6578, F3, 21) (dual of [6578, 6464, 22]-code), using
(93, 114, 557710)-Net in Base 3 — Upper bound on s
There is no (93, 114, 557711)-net in base 3, because
- 1 times m-reduction [i] would yield (93, 113, 557711)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 821680 543878 439360 521086 801490 663868 454548 986675 952397 > 3113 [i]