Best Known (78−15, 78, s)-Nets in Base 128
(78−15, 78, 1897425)-Net over F128 — Constructive and digital
Digital (63, 78, 1897425)-net over F128, using
- (u, u+v)-construction [i] based on
- digital (14, 21, 699054)-net over F128, using
- net defined by OOA [i] based on linear OOA(12821, 699054, F128, 7, 7) (dual of [(699054, 7), 4893357, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(12821, 2097163, F128, 7) (dual of [2097163, 2097142, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(12819, 2097152, F128, 7) (dual of [2097152, 2097133, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(12821, 2097163, F128, 7) (dual of [2097163, 2097142, 8]-code), using
- net defined by OOA [i] based on linear OOA(12821, 699054, F128, 7, 7) (dual of [(699054, 7), 4893357, 8]-NRT-code), using
- digital (42, 57, 1198371)-net over F128, using
- net defined by OOA [i] based on linear OOA(12857, 1198371, F128, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12857, 8388598, F128, 15) (dual of [8388598, 8388541, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(12857, large, F128, 15) (dual of [large, large−57, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 15790321 | 1288−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(12857, large, F128, 15) (dual of [large, large−57, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12857, 8388598, F128, 15) (dual of [8388598, 8388541, 16]-code), using
- net defined by OOA [i] based on linear OOA(12857, 1198371, F128, 15, 15) (dual of [(1198371, 15), 17975508, 16]-NRT-code), using
- digital (14, 21, 699054)-net over F128, using
(78−15, 78, 2396742)-Net in Base 128 — Constructive
(63, 78, 2396742)-net in base 128, using
- 1286 times duplication [i] based on (57, 72, 2396742)-net in base 128, using
- base change [i] based on digital (48, 63, 2396742)-net over F256, using
- 2561 times duplication [i] based on digital (47, 62, 2396742)-net over F256, using
- (u, u+v)-construction [i] based on
- digital (12, 19, 2796200)-net over F256, using
- net defined by OOA [i] based on linear OOA(25619, 2796200, F256, 7, 7) (dual of [(2796200, 7), 19573381, 8]-NRT-code), using
- appending kth column [i] based on linear OOA(25619, 2796200, F256, 6, 7) (dual of [(2796200, 6), 16777181, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(25619, 8388601, F256, 7) (dual of [8388601, 8388582, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(25619, large, F256, 7) (dual of [large, large−19, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(25619, 8388601, F256, 7) (dual of [8388601, 8388582, 8]-code), using
- appending kth column [i] based on linear OOA(25619, 2796200, F256, 6, 7) (dual of [(2796200, 6), 16777181, 8]-NRT-code), using
- net defined by OOA [i] based on linear OOA(25619, 2796200, F256, 7, 7) (dual of [(2796200, 7), 19573381, 8]-NRT-code), using
- digital (28, 43, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- digital (12, 19, 2796200)-net over F256, using
- (u, u+v)-construction [i] based on
- 2561 times duplication [i] based on digital (47, 62, 2396742)-net over F256, using
- base change [i] based on digital (48, 63, 2396742)-net over F256, using
(78−15, 78, large)-Net over F128 — Digital
Digital (63, 78, large)-net over F128, using
- 1281 times duplication [i] based on digital (62, 77, large)-net over F128, using
- t-expansion [i] based on digital (56, 77, large)-net over F128, using
(78−15, 78, large)-Net in Base 128 — Upper bound on s
There is no (63, 78, large)-net in base 128, because
- 13 times m-reduction [i] would yield (63, 65, large)-net in base 128, but