Best Known (76−15, 76, s)-Nets in Base 7
(76−15, 76, 16809)-Net over F7 — Constructive and digital
Digital (61, 76, 16809)-net over F7, using
- net defined by OOA [i] based on linear OOA(776, 16809, F7, 15, 15) (dual of [(16809, 15), 252059, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(776, 117664, F7, 15) (dual of [117664, 117588, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- linear OA(773, 117649, F7, 15) (dual of [117649, 117576, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(761, 117649, F7, 12) (dual of [117649, 117588, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(73, 15, F7, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(14) ⊂ Ce(11) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(776, 117664, F7, 15) (dual of [117664, 117588, 16]-code), using
(76−15, 76, 70921)-Net over F7 — Digital
Digital (61, 76, 70921)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(776, 70921, F7, 15) (dual of [70921, 70845, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(776, 117658, F7, 15) (dual of [117658, 117582, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- linear OA(773, 117650, F7, 15) (dual of [117650, 117577, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(761, 117650, F7, 11) (dual of [117650, 117589, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 117650 | 712−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(73, 8, F7, 3) (dual of [8, 5, 4]-code or 8-arc in PG(2,7) or 8-cap in PG(2,7)), using
- extended Reed–Solomon code RSe(5,7) [i]
- oval in PG(2, 7) [i]
- construction X applied to C([0,7]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(776, 117658, F7, 15) (dual of [117658, 117582, 16]-code), using
(76−15, 76, large)-Net in Base 7 — Upper bound on s
There is no (61, 76, large)-net in base 7, because
- 13 times m-reduction [i] would yield (61, 63, large)-net in base 7, but