Best Known (42, 42+8, s)-Nets in Base 9
(42, 42+8, 1195744)-Net over F9 — Constructive and digital
Digital (42, 50, 1195744)-net over F9, using
- net defined by OOA [i] based on linear OOA(950, 1195744, F9, 8, 8) (dual of [(1195744, 8), 9565902, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(950, 4782976, F9, 8) (dual of [4782976, 4782926, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(950, 4782969, F9, 8) (dual of [4782969, 4782919, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(943, 4782969, F9, 7) (dual of [4782969, 4782926, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 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
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- OA 4-folding and stacking [i] based on linear OA(950, 4782976, F9, 8) (dual of [4782976, 4782926, 9]-code), using
(42, 42+8, 4782976)-Net over F9 — Digital
Digital (42, 50, 4782976)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(950, 4782976, F9, 8) (dual of [4782976, 4782926, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(950, 4782969, F9, 8) (dual of [4782969, 4782919, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(943, 4782969, F9, 7) (dual of [4782969, 4782926, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 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
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
(42, 42+8, large)-Net in Base 9 — Upper bound on s
There is no (42, 50, large)-net in base 9, because
- 6 times m-reduction [i] would yield (42, 44, large)-net in base 9, but