Best Known (127, 140, s)-Nets in Base 8
(127, 140, 5592400)-Net over F8 — Constructive and digital
Digital (127, 140, 5592400)-net over F8, using
- trace code for nets [i] based on digital (57, 70, 2796200)-net over F64, using
- net defined by OOA [i] based on linear OOA(6470, 2796200, F64, 14, 13) (dual of [(2796200, 14), 39146730, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(6470, 8388601, F64, 2, 13) (dual of [(8388601, 2), 16777132, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(6470, 8388602, F64, 2, 13) (dual of [(8388602, 2), 16777134, 14]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(6421, 4194301, F64, 2, 6) (dual of [(4194301, 2), 8388581, 7]-NRT-code), using
- OOA 2-folding [i] based on linear OA(6421, 8388602, F64, 6) (dual of [8388602, 8388581, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(6421, large, F64, 6) (dual of [large, large−21, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(6421, large, F64, 6) (dual of [large, large−21, 7]-code), using
- OOA 2-folding [i] based on linear OA(6421, 8388602, F64, 6) (dual of [8388602, 8388581, 7]-code), using
- 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
- linear OOA(6421, 4194301, F64, 2, 6) (dual of [(4194301, 2), 8388581, 7]-NRT-code), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OOA(6470, 8388602, F64, 2, 13) (dual of [(8388602, 2), 16777134, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(6470, 8388601, F64, 2, 13) (dual of [(8388601, 2), 16777132, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(6470, 2796200, F64, 14, 13) (dual of [(2796200, 14), 39146730, 14]-NRT-code), using
(127, 140, large)-Net over F8 — Digital
Digital (127, 140, large)-net over F8, using
- t-expansion [i] based on digital (125, 140, large)-net over F8, using
- 6 times m-reduction [i] based on digital (125, 146, large)-net over F8, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8146, large, F8, 21) (dual of [large, large−146, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(8145, large, F8, 21) (dual of [large, large−145, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 816−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 1 times code embedding in larger space [i] based on linear OA(8145, large, F8, 21) (dual of [large, large−145, 22]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8146, large, F8, 21) (dual of [large, large−146, 22]-code), using
- 6 times m-reduction [i] based on digital (125, 146, large)-net over F8, using
(127, 140, large)-Net in Base 8 — Upper bound on s
There is no (127, 140, large)-net in base 8, because
- 11 times m-reduction [i] would yield (127, 129, large)-net in base 8, but