Best Known (117−19, 117, s)-Nets in Base 8
(117−19, 117, 233019)-Net over F8 — Constructive and digital
Digital (98, 117, 233019)-net over F8, using
- net defined by OOA [i] based on linear OOA(8117, 233019, F8, 19, 19) (dual of [(233019, 19), 4427244, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(8117, 2097172, F8, 19) (dual of [2097172, 2097055, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(8117, 2097177, F8, 19) (dual of [2097177, 2097060, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(8113, 2097152, F8, 19) (dual of [2097152, 2097039, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(892, 2097152, F8, 15) (dual of [2097152, 2097060, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(84, 25, F8, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,8)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(8117, 2097177, F8, 19) (dual of [2097177, 2097060, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(8117, 2097172, F8, 19) (dual of [2097172, 2097055, 20]-code), using
(117−19, 117, 1489722)-Net over F8 — Digital
Digital (98, 117, 1489722)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8117, 1489722, F8, 19) (dual of [1489722, 1489605, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(8117, 2097177, F8, 19) (dual of [2097177, 2097060, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(8113, 2097152, F8, 19) (dual of [2097152, 2097039, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(892, 2097152, F8, 15) (dual of [2097152, 2097060, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(84, 25, F8, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,8)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(8117, 2097177, F8, 19) (dual of [2097177, 2097060, 20]-code), using
(117−19, 117, large)-Net in Base 8 — Upper bound on s
There is no (98, 117, large)-net in base 8, because
- 17 times m-reduction [i] would yield (98, 100, large)-net in base 8, but