Best Known (125, 125+22, s)-Nets in Base 4
(125, 125+22, 23832)-Net over F4 — Constructive and digital
Digital (125, 147, 23832)-net over F4, using
- 42 times duplication [i] based on digital (123, 145, 23832)-net over F4, using
- net defined by OOA [i] based on linear OOA(4145, 23832, F4, 22, 22) (dual of [(23832, 22), 524159, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4145, 262152, F4, 22) (dual of [262152, 262007, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4145, 262153, F4, 22) (dual of [262153, 262008, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(4145, 262144, F4, 22) (dual of [262144, 261999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(40, 9, F4, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4145, 262153, F4, 22) (dual of [262153, 262008, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(4145, 262152, F4, 22) (dual of [262152, 262007, 23]-code), using
- net defined by OOA [i] based on linear OOA(4145, 23832, F4, 22, 22) (dual of [(23832, 22), 524159, 23]-NRT-code), using
(125, 125+22, 103920)-Net over F4 — Digital
Digital (125, 147, 103920)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4147, 103920, F4, 2, 22) (dual of [(103920, 2), 207693, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4147, 131077, F4, 2, 22) (dual of [(131077, 2), 262007, 23]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4146, 131077, F4, 2, 22) (dual of [(131077, 2), 262008, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4146, 262154, F4, 22) (dual of [262154, 262008, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4145, 262153, F4, 22) (dual of [262153, 262008, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(4145, 262144, F4, 22) (dual of [262144, 261999, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(40, 9, F4, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4145, 262153, F4, 22) (dual of [262153, 262008, 23]-code), using
- OOA 2-folding [i] based on linear OA(4146, 262154, F4, 22) (dual of [262154, 262008, 23]-code), using
- 41 times duplication [i] based on linear OOA(4146, 131077, F4, 2, 22) (dual of [(131077, 2), 262008, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4147, 131077, F4, 2, 22) (dual of [(131077, 2), 262007, 23]-NRT-code), using
(125, 125+22, large)-Net in Base 4 — Upper bound on s
There is no (125, 147, large)-net in base 4, because
- 20 times m-reduction [i] would yield (125, 127, large)-net in base 4, but