Best Known (98, 98+10, s)-Nets in Base 7
(98, 98+10, 4611844)-Net over F7 — Constructive and digital
Digital (98, 108, 4611844)-net over F7, using
- 71 times duplication [i] based on digital (97, 107, 4611844)-net over F7, using
- (u, u+v)-construction [i] based on
- digital (28, 33, 2882404)-net over F7, using
- net defined by OOA [i] based on linear OOA(733, 2882404, F7, 5, 5) (dual of [(2882404, 5), 14411987, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(733, 5764809, F7, 5) (dual of [5764809, 5764776, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(733, 5764801, F7, 5) (dual of [5764801, 5764768, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(725, 5764801, F7, 4) (dual of [5764801, 5764776, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 78−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(70, 8, F7, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(70, s, F7, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OA(733, 5764809, F7, 5) (dual of [5764809, 5764776, 6]-code), using
- net defined by OOA [i] based on linear OOA(733, 2882404, F7, 5, 5) (dual of [(2882404, 5), 14411987, 6]-NRT-code), using
- digital (64, 74, 2305922)-net over F7, using
- trace code for nets [i] based on digital (27, 37, 1152961)-net over F49, using
- net defined by OOA [i] based on linear OOA(4937, 1152961, F49, 10, 10) (dual of [(1152961, 10), 11529573, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(4937, 5764805, F49, 10) (dual of [5764805, 5764768, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(4937, 5764801, F49, 10) (dual of [5764801, 5764764, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(4933, 5764801, F49, 9) (dual of [5764801, 5764768, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 5764800 = 494−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(490, 4, F49, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(490, s, F49, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- OA 5-folding and stacking [i] based on linear OA(4937, 5764805, F49, 10) (dual of [5764805, 5764768, 11]-code), using
- net defined by OOA [i] based on linear OOA(4937, 1152961, F49, 10, 10) (dual of [(1152961, 10), 11529573, 11]-NRT-code), using
- trace code for nets [i] based on digital (27, 37, 1152961)-net over F49, using
- digital (28, 33, 2882404)-net over F7, using
- (u, u+v)-construction [i] based on
(98, 98+10, large)-Net over F7 — Digital
Digital (98, 108, large)-net over F7, using
- t-expansion [i] based on digital (94, 108, large)-net over F7, using
- 1 times m-reduction [i] based on digital (94, 109, large)-net over F7, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(7109, large, F7, 15) (dual of [large, large−109, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 20176804 | 718−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(7109, large, F7, 15) (dual of [large, large−109, 16]-code), using
- 1 times m-reduction [i] based on digital (94, 109, large)-net over F7, using
(98, 98+10, large)-Net in Base 7 — Upper bound on s
There is no (98, 108, large)-net in base 7, because
- 8 times m-reduction [i] would yield (98, 100, large)-net in base 7, but