Best Known (133−20, 133, s)-Nets in Base 9
(133−20, 133, 478325)-Net over F9 — Constructive and digital
Digital (113, 133, 478325)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (3, 13, 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 (100, 120, 478297)-net over F9, using
- net defined by OOA [i] based on linear OOA(9120, 478297, F9, 20, 20) (dual of [(478297, 20), 9565820, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(9120, 4782970, F9, 20) (dual of [4782970, 4782850, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(9120, 4782976, F9, 20) (dual of [4782976, 4782856, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(9120, 4782969, F9, 20) (dual of [4782969, 4782849, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(9113, 4782969, F9, 19) (dual of [4782969, 4782856, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(9120, 4782976, F9, 20) (dual of [4782976, 4782856, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(9120, 4782970, F9, 20) (dual of [4782970, 4782850, 21]-code), using
- net defined by OOA [i] based on linear OOA(9120, 478297, F9, 20, 20) (dual of [(478297, 20), 9565820, 21]-NRT-code), using
- digital (3, 13, 28)-net over F9, using
(133−20, 133, 4783031)-Net over F9 — Digital
Digital (113, 133, 4783031)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9133, 4783031, F9, 20) (dual of [4783031, 4782898, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(11) [i] based on
- linear OA(9120, 4782969, F9, 20) (dual of [4782969, 4782849, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(971, 4782969, F9, 12) (dual of [4782969, 4782898, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(913, 62, F9, 7) (dual of [62, 49, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(913, 80, F9, 7) (dual of [80, 67, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(913, 80, F9, 7) (dual of [80, 67, 8]-code), using
- construction X applied to Ce(19) ⊂ Ce(11) [i] based on
(133−20, 133, large)-Net in Base 9 — Upper bound on s
There is no (113, 133, large)-net in base 9, because
- 18 times m-reduction [i] would yield (113, 115, large)-net in base 9, but