Best Known (79, 79+16, s)-Nets in Base 9
(79, 79+16, 132862)-Net over F9 — Constructive and digital
Digital (79, 95, 132862)-net over F9, using
- 91 times duplication [i] based on digital (78, 94, 132862)-net over F9, using
- net defined by OOA [i] based on linear OOA(994, 132862, F9, 16, 16) (dual of [(132862, 16), 2125698, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(994, 1062896, F9, 16) (dual of [1062896, 1062802, 17]-code), using
- trace code [i] based on linear OA(8147, 531448, F81, 16) (dual of [531448, 531401, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(8146, 531441, F81, 16) (dual of [531441, 531395, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(8140, 531441, F81, 14) (dual of [531441, 531401, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- trace code [i] based on linear OA(8147, 531448, F81, 16) (dual of [531448, 531401, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(994, 1062896, F9, 16) (dual of [1062896, 1062802, 17]-code), using
- net defined by OOA [i] based on linear OOA(994, 132862, F9, 16, 16) (dual of [(132862, 16), 2125698, 17]-NRT-code), using
(79, 79+16, 1062898)-Net over F9 — Digital
Digital (79, 95, 1062898)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(995, 1062898, F9, 16) (dual of [1062898, 1062803, 17]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(994, 1062896, F9, 16) (dual of [1062896, 1062802, 17]-code), using
- trace code [i] based on linear OA(8147, 531448, F81, 16) (dual of [531448, 531401, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- linear OA(8146, 531441, F81, 16) (dual of [531441, 531395, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(8140, 531441, F81, 14) (dual of [531441, 531401, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(15) ⊂ Ce(13) [i] based on
- trace code [i] based on linear OA(8147, 531448, F81, 16) (dual of [531448, 531401, 17]-code), using
- linear OA(994, 1062897, F9, 15) (dual of [1062897, 1062803, 16]-code), using Gilbert–Varšamov bound and bm = 994 > Vbs−1(k−1) = 118 491123 120095 375458 514502 226374 386761 423750 325915 614839 347930 475070 169853 559416 209793 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(994, 1062896, F9, 16) (dual of [1062896, 1062802, 17]-code), using
- construction X with Varšamov bound [i] based on
(79, 79+16, large)-Net in Base 9 — Upper bound on s
There is no (79, 95, large)-net in base 9, because
- 14 times m-reduction [i] would yield (79, 81, large)-net in base 9, but