Best Known (89, 89+25, s)-Nets in Base 8
(89, 89+25, 2733)-Net over F8 — Constructive and digital
Digital (89, 114, 2733)-net over F8, using
- 81 times duplication [i] based on digital (88, 113, 2733)-net over F8, using
- net defined by OOA [i] based on linear OOA(8113, 2733, F8, 25, 25) (dual of [(2733, 25), 68212, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8113, 32797, F8, 25) (dual of [32797, 32684, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8113, 32800, F8, 25) (dual of [32800, 32687, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(18) [i] based on
- linear OA(8106, 32768, F8, 25) (dual of [32768, 32662, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(881, 32768, F8, 19) (dual of [32768, 32687, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(87, 32, F8, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(24) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(8113, 32800, F8, 25) (dual of [32800, 32687, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8113, 32797, F8, 25) (dual of [32797, 32684, 26]-code), using
- net defined by OOA [i] based on linear OOA(8113, 2733, F8, 25, 25) (dual of [(2733, 25), 68212, 26]-NRT-code), using
(89, 89+25, 32802)-Net over F8 — Digital
Digital (89, 114, 32802)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8114, 32802, F8, 25) (dual of [32802, 32688, 26]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8113, 32800, F8, 25) (dual of [32800, 32687, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(18) [i] based on
- linear OA(8106, 32768, F8, 25) (dual of [32768, 32662, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(881, 32768, F8, 19) (dual of [32768, 32687, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(87, 32, F8, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(24) ⊂ Ce(18) [i] based on
- linear OA(8113, 32801, F8, 24) (dual of [32801, 32688, 25]-code), using Gilbert–Varšamov bound and bm = 8113 > Vbs−1(k−1) = 77010 824640 495607 895302 842591 674104 238548 485955 522040 762847 240242 416176 920367 164619 338920 386835 360171 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(8113, 32800, F8, 25) (dual of [32800, 32687, 26]-code), using
- construction X with Varšamov bound [i] based on
(89, 89+25, large)-Net in Base 8 — Upper bound on s
There is no (89, 114, large)-net in base 8, because
- 23 times m-reduction [i] would yield (89, 91, large)-net in base 8, but