Best Known (89, 89+31, s)-Nets in Base 3
(89, 89+31, 264)-Net over F3 — Constructive and digital
Digital (89, 120, 264)-net over F3, using
- trace code for nets [i] based on digital (9, 40, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
(89, 89+31, 505)-Net over F3 — Digital
Digital (89, 120, 505)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3120, 505, F3, 31) (dual of [505, 385, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3120, 738, F3, 31) (dual of [738, 618, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(28) ⊂ Ce(27) [i] based on
- linear OA(3118, 729, F3, 31) (dual of [729, 611, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3112, 729, F3, 29) (dual of [729, 617, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3109, 729, F3, 28) (dual of [729, 620, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(31, 8, F3, 1) (dual of [8, 7, 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(30, 1, F3, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(30) ⊂ Ce(28) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3120, 738, F3, 31) (dual of [738, 618, 32]-code), using
(89, 89+31, 19569)-Net in Base 3 — Upper bound on s
There is no (89, 120, 19570)-net in base 3, because
- 1 times m-reduction [i] would yield (89, 119, 19570)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 599 288788 692021 636769 855849 359122 815413 644223 847944 935385 > 3119 [i]