Best Known (89, 89+17, s)-Nets in Base 7
(89, 89+17, 102948)-Net over F7 — Constructive and digital
Digital (89, 106, 102948)-net over F7, using
- net defined by OOA [i] based on linear OOA(7106, 102948, F7, 17, 17) (dual of [(102948, 17), 1750010, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(7106, 823585, F7, 17) (dual of [823585, 823479, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(799, 823543, F7, 17) (dual of [823543, 823444, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(764, 823543, F7, 11) (dual of [823543, 823479, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(77, 42, F7, 5) (dual of [42, 35, 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
- OOA 8-folding and stacking with additional row [i] based on linear OA(7106, 823585, F7, 17) (dual of [823585, 823479, 18]-code), using
(89, 89+17, 823585)-Net over F7 — Digital
Digital (89, 106, 823585)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7106, 823585, F7, 17) (dual of [823585, 823479, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(10) [i] based on
- linear OA(799, 823543, F7, 17) (dual of [823543, 823444, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(764, 823543, F7, 11) (dual of [823543, 823479, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(77, 42, F7, 5) (dual of [42, 35, 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
(89, 89+17, large)-Net in Base 7 — Upper bound on s
There is no (89, 106, large)-net in base 7, because
- 15 times m-reduction [i] would yield (89, 91, large)-net in base 7, but