Best Known (238−36, 238, s)-Nets in Base 3
(238−36, 238, 1500)-Net over F3 — Constructive and digital
Digital (202, 238, 1500)-net over F3, using
- 31 times duplication [i] based on digital (201, 237, 1500)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (11, 29, 20)-net over F3, using
- net from sequence [i] based on digital (11, 19)-sequence over F3, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F3 with g(F) = 9, N(F) = 19, and 1 place with degree 3 [i] based on function field F/F3 with g(F) = 9 and N(F) ≥ 19, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (11, 19)-sequence over F3, using
- digital (172, 208, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 52, 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, 52, 370)-net over F81, using
- digital (11, 29, 20)-net over F3, using
- (u, u+v)-construction [i] based on
(238−36, 238, 14298)-Net over F3 — Digital
Digital (202, 238, 14298)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3238, 14298, F3, 36) (dual of [14298, 14060, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(3238, 19759, F3, 36) (dual of [19759, 19521, 37]-code), using
- construction X applied to C([0,18]) ⊂ C([0,13]) [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(3163, 19684, F3, 27) (dual of [19684, 19521, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(321, 75, F3, 8) (dual of [75, 54, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(321, 80, F3, 8) (dual of [80, 59, 9]-code), using
- construction X applied to C([0,18]) ⊂ C([0,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3238, 19759, F3, 36) (dual of [19759, 19521, 37]-code), using
(238−36, 238, 7686034)-Net in Base 3 — Upper bound on s
There is no (202, 238, 7686035)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 358805 540563 351739 808834 699406 318773 388381 711928 578392 823135 828197 126081 359457 581376 363498 715677 538697 058669 711781 > 3238 [i]