Best Known (88, 115, s)-Nets in Base 3
(88, 115, 400)-Net over F3 — Constructive and digital
Digital (88, 115, 400)-net over F3, using
- 1 times m-reduction [i] based on digital (88, 116, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 29, 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
- trace code for nets [i] based on digital (1, 29, 100)-net over F81, using
(88, 115, 740)-Net over F3 — Digital
Digital (88, 115, 740)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3115, 740, F3, 27) (dual of [740, 625, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3115, 758, F3, 27) (dual of [758, 643, 28]-code), using
- construction XX applied to C1 = C([724,21]), C2 = C([0,22]), C3 = C1 + C2 = C([0,21]), and C∩ = C1 ∩ C2 = C([724,22]) [i] based on
- linear OA(3103, 728, F3, 26) (dual of [728, 625, 27]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,21}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(391, 728, F3, 23) (dual of [728, 637, 24]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(3109, 728, F3, 27) (dual of [728, 619, 28]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−4,−3,…,22}, and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(385, 728, F3, 22) (dual of [728, 643, 23]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [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([724,21]), C2 = C([0,22]), C3 = C1 + C2 = C([0,21]), and C∩ = C1 ∩ C2 = C([724,22]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3115, 758, F3, 27) (dual of [758, 643, 28]-code), using
(88, 115, 43275)-Net in Base 3 — Upper bound on s
There is no (88, 115, 43276)-net in base 3, because
- 1 times m-reduction [i] would yield (88, 114, 43276)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2 465769 145899 331769 590150 664252 413704 543507 014487 201273 > 3114 [i]