Best Known (135−13, 135, s)-Nets in Base 8
(135−13, 135, 3495256)-Net over F8 — Constructive and digital
Digital (122, 135, 3495256)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (31, 37, 699056)-net over F8, using
- net defined by OOA [i] based on linear OOA(837, 699056, F8, 6, 6) (dual of [(699056, 6), 4194299, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(837, 2097168, F8, 6) (dual of [2097168, 2097131, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(836, 2097152, F8, 6) (dual of [2097152, 2097116, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(822, 2097152, F8, 4) (dual of [2097152, 2097130, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(815, 16, F8, 15) (dual of [16, 1, 16]-code or 16-arc in PG(14,8)), using
- dual of repetition code with length 16 [i]
- linear OA(81, 16, F8, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(837, 2097168, F8, 6) (dual of [2097168, 2097131, 7]-code), using
- net defined by OOA [i] based on linear OOA(837, 699056, F8, 6, 6) (dual of [(699056, 6), 4194299, 7]-NRT-code), using
- digital (85, 98, 2796200)-net over F8, using
- net defined by OOA [i] based on linear OOA(898, 2796200, F8, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(898, 8388601, F8, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(898, 8388602, F8, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- trace code [i] based on linear OOA(6449, 4194301, F64, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6449, 8388602, F64, 13) (dual of [8388602, 8388553, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- OOA 2-folding [i] based on linear OA(6449, 8388602, F64, 13) (dual of [8388602, 8388553, 14]-code), using
- trace code [i] based on linear OOA(6449, 4194301, F64, 2, 13) (dual of [(4194301, 2), 8388553, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(898, 8388602, F8, 2, 13) (dual of [(8388602, 2), 16777106, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(898, 8388601, F8, 2, 13) (dual of [(8388601, 2), 16777104, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(898, 2796200, F8, 14, 13) (dual of [(2796200, 14), 39146702, 14]-NRT-code), using
- digital (31, 37, 699056)-net over F8, using
(135−13, 135, large)-Net over F8 — Digital
Digital (122, 135, large)-net over F8, using
- t-expansion [i] based on digital (119, 135, large)-net over F8, using
- 4 times m-reduction [i] based on digital (119, 139, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8139, large, F8, 20) (dual of [large, large−139, 21]-code), using
- 2 times code embedding in larger space [i] based on linear OA(8137, large, F8, 20) (dual of [large, large−137, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 88−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 2 times code embedding in larger space [i] based on linear OA(8137, large, F8, 20) (dual of [large, large−137, 21]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8139, large, F8, 20) (dual of [large, large−139, 21]-code), using
- 4 times m-reduction [i] based on digital (119, 139, large)-net over F8, using
(135−13, 135, large)-Net in Base 8 — Upper bound on s
There is no (122, 135, large)-net in base 8, because
- 11 times m-reduction [i] would yield (122, 124, large)-net in base 8, but