Best Known (83, 115, s)-Nets in Base 9
(83, 115, 768)-Net over F9 — Constructive and digital
Digital (83, 115, 768)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (3, 19, 28)-net over F9, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- the Hermitian function field over F9 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 3 and N(F) ≥ 28, using
- net from sequence [i] based on digital (3, 27)-sequence over F9, using
- digital (64, 96, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 48, 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, 48, 370)-net over F81, using
- digital (3, 19, 28)-net over F9, using
(83, 115, 6348)-Net over F9 — Digital
Digital (83, 115, 6348)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9115, 6348, F9, 32) (dual of [6348, 6233, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(9115, 6572, F9, 32) (dual of [6572, 6457, 33]-code), using
- construction XX applied to Ce(31) ⊂ Ce(29) ⊂ Ce(28) [i] based on
- linear OA(9113, 6561, F9, 32) (dual of [6561, 6448, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(9105, 6561, F9, 30) (dual of [6561, 6456, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(9101, 6561, F9, 29) (dual of [6561, 6460, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(91, 10, F9, 1) (dual of [10, 9, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 93−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(91, 728, F9, 1) (dual of [728, 727, 2]-code), using
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(31) ⊂ Ce(29) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(9115, 6572, F9, 32) (dual of [6572, 6457, 33]-code), using
(83, 115, 6138554)-Net in Base 9 — Upper bound on s
There is no (83, 115, 6138555)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 54 687664 135091 948175 968818 853366 845013 242721 019018 931012 871038 733390 780268 215829 468553 018963 002345 588768 917889 > 9115 [i]