Best Known (118−21, 118, s)-Nets in Base 9
(118−21, 118, 53147)-Net over F9 — Constructive and digital
Digital (97, 118, 53147)-net over F9, using
- 93 times duplication [i] based on digital (94, 115, 53147)-net over F9, using
- net defined by OOA [i] based on linear OOA(9115, 53147, F9, 21, 21) (dual of [(53147, 21), 1115972, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(9115, 531471, F9, 21) (dual of [531471, 531356, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(9109, 531441, F9, 21) (dual of [531441, 531332, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(985, 531441, F9, 16) (dual of [531441, 531356, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(96, 30, F9, 4) (dual of [30, 24, 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
- OOA 10-folding and stacking with additional row [i] based on linear OA(9115, 531471, F9, 21) (dual of [531471, 531356, 22]-code), using
- net defined by OOA [i] based on linear OOA(9115, 53147, F9, 21, 21) (dual of [(53147, 21), 1115972, 22]-NRT-code), using
(118−21, 118, 531482)-Net over F9 — Digital
Digital (97, 118, 531482)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9118, 531482, F9, 21) (dual of [531482, 531364, 22]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9116, 531478, F9, 21) (dual of [531478, 531362, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(9109, 531441, F9, 21) (dual of [531441, 531332, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(979, 531441, F9, 15) (dual of [531441, 531362, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(97, 37, F9, 5) (dual of [37, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(20) ⊂ Ce(14) [i] based on
- linear OA(9116, 531480, F9, 20) (dual of [531480, 531364, 21]-code), using Gilbert–Varšamov bound and bm = 9116 > Vbs−1(k−1) = 7 206317 853571 147259 324326 736842 662446 122448 018094 642206 782415 107987 926868 684916 623369 761556 302344 119166 330361 [i]
- linear OA(90, 2, F9, 0) (dual of [2, 2, 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
- linear OA(9116, 531478, F9, 21) (dual of [531478, 531362, 22]-code), using
- construction X with Varšamov bound [i] based on
(118−21, 118, large)-Net in Base 9 — Upper bound on s
There is no (97, 118, large)-net in base 9, because
- 19 times m-reduction [i] would yield (97, 99, large)-net in base 9, but