Best Known (198−31, 198, s)-Nets in Base 3
(198−31, 198, 1480)-Net over F3 — Constructive and digital
Digital (167, 198, 1480)-net over F3, using
- t-expansion [i] based on digital (166, 198, 1480)-net over F3, using
- 2 times m-reduction [i] based on digital (166, 200, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 50, 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, 50, 370)-net over F81, using
- 2 times m-reduction [i] based on digital (166, 200, 1480)-net over F3, using
(198−31, 198, 10140)-Net over F3 — Digital
Digital (167, 198, 10140)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3198, 10140, F3, 31) (dual of [10140, 9942, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3198, 19745, F3, 31) (dual of [19745, 19547, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- linear OA(3181, 19683, F3, 31) (dual of [19683, 19502, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3136, 19683, F3, 23) (dual of [19683, 19547, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(317, 62, F3, 7) (dual of [62, 45, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(317, 80, F3, 7) (dual of [80, 63, 8]-code), using
- construction X applied to Ce(30) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3198, 19745, F3, 31) (dual of [19745, 19547, 32]-code), using
(198−31, 198, 5928271)-Net in Base 3 — Upper bound on s
There is no (167, 198, 5928272)-net in base 3, because
- 1 times m-reduction [i] would yield (167, 197, 5928272)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 9837 575587 476283 049467 223293 552357 559108 017026 849907 726589 959561 099238 039797 937532 125357 302721 > 3197 [i]