Best Known (58, 58+21, s)-Nets in Base 9
(58, 58+21, 740)-Net over F9 — Constructive and digital
Digital (58, 79, 740)-net over F9, using
- 5 times m-reduction [i] based on digital (58, 84, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 42, 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, 42, 370)-net over F81, using
(58, 58+21, 6583)-Net over F9 — Digital
Digital (58, 79, 6583)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(979, 6583, F9, 21) (dual of [6583, 6504, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(973, 6561, F9, 21) (dual of [6561, 6488, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(957, 6561, F9, 16) (dual of [6561, 6504, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(96, 22, F9, 4) (dual of [22, 16, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- 1 times truncation [i] based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
(58, 58+21, large)-Net in Base 9 — Upper bound on s
There is no (58, 79, large)-net in base 9, because
- 19 times m-reduction [i] would yield (58, 60, large)-net in base 9, but