Best Known (61, 78, s)-Nets in Base 7
(61, 78, 2104)-Net over F7 — Constructive and digital
Digital (61, 78, 2104)-net over F7, using
- 71 times duplication [i] based on digital (60, 77, 2104)-net over F7, using
- net defined by OOA [i] based on linear OOA(777, 2104, F7, 17, 17) (dual of [(2104, 17), 35691, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(777, 16833, F7, 17) (dual of [16833, 16756, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- linear OA(771, 16807, F7, 17) (dual of [16807, 16736, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(76, 26, F7, 4) (dual of [26, 20, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- 1 times truncation [i] based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(76, 42, F7, 4) (dual of [42, 36, 5]-code), using
- construction X applied to Ce(16) ⊂ Ce(11) [i] based on
- OOA 8-folding and stacking with additional row [i] based on linear OA(777, 16833, F7, 17) (dual of [16833, 16756, 18]-code), using
- net defined by OOA [i] based on linear OOA(777, 2104, F7, 17, 17) (dual of [(2104, 17), 35691, 18]-NRT-code), using
(61, 78, 16839)-Net over F7 — Digital
Digital (61, 78, 16839)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(778, 16839, F7, 17) (dual of [16839, 16761, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(771, 16807, F7, 17) (dual of [16807, 16736, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(746, 16807, F7, 11) (dual of [16807, 16761, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(77, 32, F7, 5) (dual of [32, 25, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(77, 43, F7, 5) (dual of [43, 36, 6]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
(61, 78, large)-Net in Base 7 — Upper bound on s
There is no (61, 78, large)-net in base 7, because
- 15 times m-reduction [i] would yield (61, 63, large)-net in base 7, but