Best Known (122−34, 122, s)-Nets in Base 9
(122−34, 122, 772)-Net over F9 — Constructive and digital
Digital (88, 122, 772)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (5, 22, 32)-net over F9, using
- net from sequence [i] based on digital (5, 31)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 5 and N(F) ≥ 32, using
- net from sequence [i] based on digital (5, 31)-sequence over F9, using
- digital (66, 100, 740)-net over F9, 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
- digital (5, 22, 32)-net over F9, using
(122−34, 122, 6468)-Net over F9 — Digital
Digital (88, 122, 6468)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9122, 6468, F9, 34) (dual of [6468, 6346, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(9122, 6570, F9, 34) (dual of [6570, 6448, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- linear OA(9121, 6561, F9, 34) (dual of [6561, 6440, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- 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(91, 9, F9, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(33) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(9122, 6570, F9, 34) (dual of [6570, 6448, 35]-code), using
(122−34, 122, 6323358)-Net in Base 9 — Upper bound on s
There is no (88, 122, 6323359)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 261 569045 830930 441588 693771 127239 671384 526161 066577 640623 991840 485928 696650 969845 329414 256714 755647 230657 063670 682745 > 9122 [i]