Best Known (125−20, 125, s)-Nets in Base 9
(125−20, 125, 478299)-Net over F9 — Constructive and digital
Digital (105, 125, 478299)-net over F9, using
- 91 times duplication [i] based on digital (104, 124, 478299)-net over F9, using
- net defined by OOA [i] based on linear OOA(9124, 478299, F9, 20, 20) (dual of [(478299, 20), 9565856, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(9124, 4782990, F9, 20) (dual of [4782990, 4782866, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(9124, 4782994, F9, 20) (dual of [4782994, 4782870, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(15) [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(999, 4782969, F9, 16) (dual of [4782969, 4782870, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(94, 25, F9, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,9)), using
- construction X applied to Ce(19) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(9124, 4782994, F9, 20) (dual of [4782994, 4782870, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(9124, 4782990, F9, 20) (dual of [4782990, 4782866, 21]-code), using
- net defined by OOA [i] based on linear OOA(9124, 478299, F9, 20, 20) (dual of [(478299, 20), 9565856, 21]-NRT-code), using
(125−20, 125, 3537606)-Net over F9 — Digital
Digital (105, 125, 3537606)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9125, 3537606, F9, 20) (dual of [3537606, 3537481, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(9125, 4782996, F9, 20) (dual of [4782996, 4782871, 21]-code), using
- construction XX applied to Ce(19) ⊂ Ce(15) ⊂ Ce(14) [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(999, 4782969, F9, 16) (dual of [4782969, 4782870, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(992, 4782969, F9, 15) (dual of [4782969, 4782877, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(94, 26, F9, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,9)), 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(19) ⊂ Ce(15) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(9125, 4782996, F9, 20) (dual of [4782996, 4782871, 21]-code), using
(125−20, 125, large)-Net in Base 9 — Upper bound on s
There is no (105, 125, large)-net in base 9, because
- 18 times m-reduction [i] would yield (105, 107, large)-net in base 9, but