Best Known (87, 87+17, s)-Nets in Base 7
(87, 87+17, 102946)-Net over F7 — Constructive and digital
Digital (87, 104, 102946)-net over F7, using
- net defined by OOA [i] based on linear OOA(7104, 102946, F7, 17, 17) (dual of [(102946, 17), 1749978, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(7104, 823569, F7, 17) (dual of [823569, 823465, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(7104, 823570, F7, 17) (dual of [823570, 823466, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(11) [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(778, 823543, F7, 13) (dual of [823543, 823465, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(771, 823543, F7, 12) (dual of [823543, 823472, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(74, 26, F7, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,7)), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(7104, 823570, F7, 17) (dual of [823570, 823466, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(7104, 823569, F7, 17) (dual of [823569, 823465, 18]-code), using
(87, 87+17, 680168)-Net over F7 — Digital
Digital (87, 104, 680168)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(7104, 680168, F7, 17) (dual of [680168, 680064, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(7104, 823570, F7, 17) (dual of [823570, 823466, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(11) [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(778, 823543, F7, 13) (dual of [823543, 823465, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(771, 823543, F7, 12) (dual of [823543, 823472, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 823542 = 77−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(74, 26, F7, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,7)), using
- linear OA(70, 1, F7, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(7104, 823570, F7, 17) (dual of [823570, 823466, 18]-code), using
(87, 87+17, large)-Net in Base 7 — Upper bound on s
There is no (87, 104, large)-net in base 7, because
- 15 times m-reduction [i] would yield (87, 89, large)-net in base 7, but