Best Known (132−21, 132, s)-Nets in Base 9
(132−21, 132, 478299)-Net over F9 — Constructive and digital
Digital (111, 132, 478299)-net over F9, using
- 91 times duplication [i] based on digital (110, 131, 478299)-net over F9, using
- net defined by OOA [i] based on linear OOA(9131, 478299, F9, 21, 21) (dual of [(478299, 21), 10044148, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(9131, 4782991, F9, 21) (dual of [4782991, 4782860, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(9131, 4782994, F9, 21) (dual of [4782994, 4782863, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(9127, 4782969, F9, 21) (dual of [4782969, 4782842, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(9106, 4782969, F9, 17) (dual of [4782969, 4782863, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(94, 25, F9, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,9)), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(9131, 4782994, F9, 21) (dual of [4782994, 4782863, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(9131, 4782991, F9, 21) (dual of [4782991, 4782860, 22]-code), using
- net defined by OOA [i] based on linear OOA(9131, 478299, F9, 21, 21) (dual of [(478299, 21), 10044148, 22]-NRT-code), using
(132−21, 132, 3761623)-Net over F9 — Digital
Digital (111, 132, 3761623)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9132, 3761623, F9, 21) (dual of [3761623, 3761491, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(9132, 4782996, F9, 21) (dual of [4782996, 4782864, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- linear OA(9127, 4782969, F9, 21) (dual of [4782969, 4782842, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(9106, 4782969, F9, 17) (dual of [4782969, 4782863, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(999, 4782969, F9, 16) (dual of [4782969, 4782870, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(94, 26, F9, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,9)), using
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(9132, 4782996, F9, 21) (dual of [4782996, 4782864, 22]-code), using
(132−21, 132, large)-Net in Base 9 — Upper bound on s
There is no (111, 132, large)-net in base 9, because
- 19 times m-reduction [i] would yield (111, 113, large)-net in base 9, but