Best Known (101, 101+12, s)-Nets in Base 8
(101, 101+12, 2798932)-Net over F8 — Constructive and digital
Digital (101, 113, 2798932)-net over F8, using
- 81 times duplication [i] based on digital (100, 112, 2798932)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (16, 22, 2732)-net over F8, using
- net defined by OOA [i] based on linear OOA(822, 2732, F8, 6, 6) (dual of [(2732, 6), 16370, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(822, 8196, F8, 6) (dual of [8196, 8174, 7]-code), using
- trace code [i] based on linear OA(6411, 4098, F64, 6) (dual of [4098, 4087, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(6411, 4096, F64, 6) (dual of [4096, 4085, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(649, 4096, F64, 5) (dual of [4096, 4087, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- trace code [i] based on linear OA(6411, 4098, F64, 6) (dual of [4098, 4087, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(822, 8196, F8, 6) (dual of [8196, 8174, 7]-code), using
- net defined by OOA [i] based on linear OOA(822, 2732, F8, 6, 6) (dual of [(2732, 6), 16370, 7]-NRT-code), using
- digital (78, 90, 2796200)-net over F8, using
- net defined by OOA [i] based on linear OOA(890, 2796200, F8, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(890, 8388601, F8, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(890, 8388602, F8, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
- trace code [i] based on linear OOA(6445, 4194301, F64, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6445, 8388602, F64, 12) (dual of [8388602, 8388557, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- OOA 2-folding [i] based on linear OA(6445, 8388602, F64, 12) (dual of [8388602, 8388557, 13]-code), using
- trace code [i] based on linear OOA(6445, 4194301, F64, 2, 12) (dual of [(4194301, 2), 8388557, 13]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(890, 8388602, F8, 2, 12) (dual of [(8388602, 2), 16777114, 13]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(890, 8388601, F8, 2, 12) (dual of [(8388601, 2), 16777112, 13]-NRT-code), using
- net defined by OOA [i] based on linear OOA(890, 2796200, F8, 14, 12) (dual of [(2796200, 14), 39146710, 13]-NRT-code), using
- digital (16, 22, 2732)-net over F8, using
- (u, u+v)-construction [i] based on
(101, 101+12, large)-Net over F8 — Digital
Digital (101, 113, large)-net over F8, using
- t-expansion [i] based on digital (100, 113, large)-net over F8, using
- 4 times m-reduction [i] based on digital (100, 117, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8117, large, F8, 17) (dual of [large, large−117, 18]-code), using
- 4 times code embedding in larger space [i] based on linear OA(8113, large, F8, 17) (dual of [large, large−113, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 816−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- 4 times code embedding in larger space [i] based on linear OA(8113, large, F8, 17) (dual of [large, large−113, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8117, large, F8, 17) (dual of [large, large−117, 18]-code), using
- 4 times m-reduction [i] based on digital (100, 117, large)-net over F8, using
(101, 101+12, large)-Net in Base 8 — Upper bound on s
There is no (101, 113, large)-net in base 8, because
- 10 times m-reduction [i] would yield (101, 103, large)-net in base 8, but