Best Known (54−12, 54, s)-Nets in Base 3
(54−12, 54, 400)-Net over F3 — Constructive and digital
Digital (42, 54, 400)-net over F3, using
- 32 times duplication [i] based on digital (40, 52, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 13, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 13, 100)-net over F81, using
(54−12, 54, 756)-Net over F3 — Digital
Digital (42, 54, 756)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(354, 756, F3, 12) (dual of [756, 702, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(354, 757, F3, 12) (dual of [757, 703, 13]-code), using
- construction XX applied to C1 = C([725,6]), C2 = C([0,9]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([725,9]) [i] based on
- linear OA(337, 728, F3, 10) (dual of [728, 691, 11]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,6}, and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(337, 728, F3, 10) (dual of [728, 691, 11]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(349, 728, F3, 13) (dual of [728, 679, 14]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−3,−2,…,9}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(325, 728, F3, 7) (dual of [728, 703, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 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
- linear OA(34, 16, F3, 2) (dual of [16, 12, 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 XX applied to C1 = C([725,6]), C2 = C([0,9]), C3 = C1 + C2 = C([0,6]), and C∩ = C1 ∩ C2 = C([725,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(354, 757, F3, 12) (dual of [757, 703, 13]-code), using
(54−12, 54, 29457)-Net in Base 3 — Upper bound on s
There is no (42, 54, 29458)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 58 150500 174458 180176 034205 > 354 [i]