Best Known (62−14, 62, s)-Nets in Base 7
(62−14, 62, 2402)-Net over F7 — Constructive and digital
Digital (48, 62, 2402)-net over F7, using
- net defined by OOA [i] based on linear OOA(762, 2402, F7, 14, 14) (dual of [(2402, 14), 33566, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(762, 16814, F7, 14) (dual of [16814, 16752, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(762, 16818, F7, 14) (dual of [16818, 16756, 15]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(761, 16807, F7, 15) (dual of [16807, 16746, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(751, 16807, F7, 12) (dual of [16807, 16756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(762, 16818, F7, 14) (dual of [16818, 16756, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(762, 16814, F7, 14) (dual of [16814, 16752, 15]-code), using
(62−14, 62, 16818)-Net over F7 — Digital
Digital (48, 62, 16818)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(762, 16818, F7, 14) (dual of [16818, 16756, 15]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(761, 16807, F7, 15) (dual of [16807, 16746, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(751, 16807, F7, 12) (dual of [16807, 16756, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(71, 11, F7, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
(62−14, 62, large)-Net in Base 7 — Upper bound on s
There is no (48, 62, large)-net in base 7, because
- 12 times m-reduction [i] would yield (48, 50, large)-net in base 7, but