Best Known (113−18, 113, s)-Nets in Base 9
(113−18, 113, 531441)-Net over F9 — Constructive and digital
Digital (95, 113, 531441)-net over F9, using
- t-expansion [i] based on digital (94, 113, 531441)-net over F9, using
- net defined by OOA [i] based on linear OOA(9113, 531441, F9, 19, 19) (dual of [(531441, 19), 10097266, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(9113, 4782970, F9, 19) (dual of [4782970, 4782857, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 4782970 | 914−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(9113, 4782970, F9, 19) (dual of [4782970, 4782857, 20]-code), using
- net defined by OOA [i] based on linear OOA(9113, 531441, F9, 19, 19) (dual of [(531441, 19), 10097266, 20]-NRT-code), using
(113−18, 113, 4065794)-Net over F9 — Digital
Digital (95, 113, 4065794)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9113, 4065794, F9, 18) (dual of [4065794, 4065681, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(9113, 4782976, F9, 18) (dual of [4782976, 4782863, 19]-code), using
- 1 times truncation [i] based on linear OA(9114, 4782977, F9, 19) (dual of [4782977, 4782863, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(16) [i] based on
- 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(9106, 4782969, F9, 17) (dual of [4782969, 4782863, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(91, 8, F9, 1) (dual of [8, 7, 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(18) ⊂ Ce(16) [i] based on
- 1 times truncation [i] based on linear OA(9114, 4782977, F9, 19) (dual of [4782977, 4782863, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(9113, 4782976, F9, 18) (dual of [4782976, 4782863, 19]-code), using
(113−18, 113, large)-Net in Base 9 — Upper bound on s
There is no (95, 113, large)-net in base 9, because
- 16 times m-reduction [i] would yield (95, 97, large)-net in base 9, but