Best Known (59, 72, s)-Nets in Base 7
(59, 72, 19613)-Net over F7 — Constructive and digital
Digital (59, 72, 19613)-net over F7, using
- net defined by OOA [i] based on linear OOA(772, 19613, F7, 13, 13) (dual of [(19613, 13), 254897, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(772, 117679, F7, 13) (dual of [117679, 117607, 14]-code), using
- construction XX applied to Ce(12) ⊂ Ce(8) ⊂ Ce(7) [i] based on
- linear OA(767, 117649, F7, 13) (dual of [117649, 117582, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(743, 117649, F7, 9) (dual of [117649, 117606, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(74, 29, F7, 3) (dual of [29, 25, 4]-code or 29-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(12) ⊂ Ce(8) ⊂ Ce(7) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(772, 117679, F7, 13) (dual of [117679, 117607, 14]-code), using
(59, 72, 117679)-Net over F7 — Digital
Digital (59, 72, 117679)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(772, 117679, F7, 13) (dual of [117679, 117607, 14]-code), using
- construction XX applied to Ce(12) ⊂ Ce(8) ⊂ Ce(7) [i] based on
- linear OA(767, 117649, F7, 13) (dual of [117649, 117582, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(743, 117649, F7, 9) (dual of [117649, 117606, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(737, 117649, F7, 8) (dual of [117649, 117612, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(74, 29, F7, 3) (dual of [29, 25, 4]-code or 29-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(12) ⊂ Ce(8) ⊂ Ce(7) [i] based on
(59, 72, large)-Net in Base 7 — Upper bound on s
There is no (59, 72, large)-net in base 7, because
- 11 times m-reduction [i] would yield (59, 61, large)-net in base 7, but