Best Known (148−36, 148, s)-Nets in Base 3
(148−36, 148, 400)-Net over F3 — Constructive and digital
Digital (112, 148, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 37, 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
(148−36, 148, 752)-Net over F3 — Digital
Digital (112, 148, 752)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3148, 752, F3, 36) (dual of [752, 604, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3148, 758, F3, 36) (dual of [758, 610, 37]-code), using
- construction XX applied to C1 = C([724,30]), C2 = C([0,31]), C3 = C1 + C2 = C([0,30]), and C∩ = C1 ∩ C2 = C([724,31]) [i] based on
- linear OA(3136, 728, F3, 35) (dual of [728, 592, 36]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,30}, and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(3124, 728, F3, 32) (dual of [728, 604, 33]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,31], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3142, 728, F3, 36) (dual of [728, 586, 37]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,31}, and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(3118, 728, F3, 31) (dual of [728, 610, 32]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,30], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(36, 24, F3, 3) (dual of [24, 18, 4]-code or 24-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
- linear OA(30, 6, F3, 0) (dual of [6, 6, 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 XX applied to C1 = C([724,30]), C2 = C([0,31]), C3 = C1 + C2 = C([0,30]), and C∩ = C1 ∩ C2 = C([724,31]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3148, 758, F3, 36) (dual of [758, 610, 37]-code), using
(148−36, 148, 31612)-Net in Base 3 — Upper bound on s
There is no (112, 148, 31613)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 41125 096297 703818 411447 379122 851326 509433 082807 151570 758493 770151 909441 > 3148 [i]