Best Known (105−24, 105, s)-Nets in Base 3
(105−24, 105, 464)-Net over F3 — Constructive and digital
Digital (81, 105, 464)-net over F3, using
- 31 times duplication [i] based on digital (80, 104, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 26, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 26, 116)-net over F81, using
(105−24, 105, 779)-Net over F3 — Digital
Digital (81, 105, 779)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3105, 779, F3, 24) (dual of [779, 674, 25]-code), using
- 31 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 11 times 0) [i] based on linear OA(397, 740, F3, 24) (dual of [740, 643, 25]-code), using
- construction XX applied to C1 = C([727,21]), C2 = C([0,22]), C3 = C1 + C2 = C([0,21]), and C∩ = C1 ∩ C2 = C([727,22]) [i] based on
- linear OA(391, 728, F3, 23) (dual of [728, 637, 24]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,…,21}, and designed minimum distance d ≥ |I|+1 = 24 [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(397, 728, F3, 24) (dual of [728, 631, 25]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,…,22}, and designed minimum distance d ≥ |I|+1 = 25 [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(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
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code) (see above)
- construction XX applied to C1 = C([727,21]), C2 = C([0,22]), C3 = C1 + C2 = C([0,21]), and C∩ = C1 ∩ C2 = C([727,22]) [i] based on
- 31 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 11 times 0) [i] based on linear OA(397, 740, F3, 24) (dual of [740, 643, 25]-code), using
(105−24, 105, 39538)-Net in Base 3 — Upper bound on s
There is no (81, 105, 39539)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 125 269844 439372 155106 558992 553728 217531 522474 434889 > 3105 [i]