Best Known (232−37, 232, s)-Nets in Base 3
(232−37, 232, 1488)-Net over F3 — Constructive and digital
Digital (195, 232, 1488)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (2, 20, 8)-net over F3, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 2 and N(F) ≥ 8, using
- net from sequence [i] based on digital (2, 7)-sequence over F3, using
- digital (175, 212, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 53, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 53, 370)-net over F81, using
- digital (2, 20, 8)-net over F3, using
(232−37, 232, 9867)-Net over F3 — Digital
Digital (195, 232, 9867)-net over F3, using
- 31 times duplication [i] based on digital (194, 231, 9867)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3231, 9867, F3, 2, 37) (dual of [(9867, 2), 19503, 38]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3229, 9866, F3, 2, 37) (dual of [(9866, 2), 19503, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3229, 19732, F3, 37) (dual of [19732, 19503, 38]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- linear OA(3217, 19684, F3, 37) (dual of [19684, 19467, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(3181, 19684, F3, 31) (dual of [19684, 19503, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(312, 48, F3, 5) (dual of [48, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- 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(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(312, 54, F3, 5) (dual of [54, 42, 6]-code), using
- construction X applied to C([0,18]) ⊂ C([0,15]) [i] based on
- OOA 2-folding [i] based on linear OA(3229, 19732, F3, 37) (dual of [19732, 19503, 38]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3229, 9866, F3, 2, 37) (dual of [(9866, 2), 19503, 38]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3231, 9867, F3, 2, 37) (dual of [(9867, 2), 19503, 38]-NRT-code), using
(232−37, 232, 5013657)-Net in Base 3 — Upper bound on s
There is no (195, 232, 5013658)-net in base 3, because
- 1 times m-reduction [i] would yield (195, 231, 5013658)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 164 063092 028245 900859 874522 965969 851118 444259 991232 097691 860659 507559 211411 464536 227413 183771 615351 407991 534981 > 3231 [i]