Best Known (63, 87, s)-Nets in Base 9
(63, 87, 740)-Net over F9 — Constructive and digital
Digital (63, 87, 740)-net over F9, using
- 7 times m-reduction [i] based on digital (63, 94, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 47, 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, 47, 370)-net over F81, using
(63, 87, 6068)-Net over F9 — Digital
Digital (63, 87, 6068)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(987, 6068, F9, 24) (dual of [6068, 5981, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(987, 6572, F9, 24) (dual of [6572, 6485, 25]-code), using
- construction XX applied to Ce(23) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- linear OA(985, 6561, F9, 24) (dual of [6561, 6476, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(977, 6561, F9, 22) (dual of [6561, 6484, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 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(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(23) ⊂ Ce(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(987, 6572, F9, 24) (dual of [6572, 6485, 25]-code), using
(63, 87, 5476827)-Net in Base 9 — Upper bound on s
There is no (63, 87, 5476828)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 104495 783357 055583 274449 134573 256353 766656 292863 639230 172263 877347 469326 198420 364673 > 987 [i]