Best Known (93, 93+22, s)-Nets in Base 4
(93, 93+22, 1490)-Net over F4 — Constructive and digital
Digital (93, 115, 1490)-net over F4, using
- 42 times duplication [i] based on digital (91, 113, 1490)-net over F4, using
- net defined by OOA [i] based on linear OOA(4113, 1490, F4, 22, 22) (dual of [(1490, 22), 32667, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4113, 16390, F4, 22) (dual of [16390, 16277, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4113, 16391, F4, 22) (dual of [16391, 16278, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 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(4113, 16391, F4, 22) (dual of [16391, 16278, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(4113, 16390, F4, 22) (dual of [16390, 16277, 23]-code), using
- net defined by OOA [i] based on linear OOA(4113, 1490, F4, 22, 22) (dual of [(1490, 22), 32667, 23]-NRT-code), using
(93, 93+22, 8197)-Net over F4 — Digital
Digital (93, 115, 8197)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4115, 8197, F4, 2, 22) (dual of [(8197, 2), 16279, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4115, 16394, F4, 22) (dual of [16394, 16279, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(499, 16384, F4, 19) (dual of [16384, 16285, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 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
- linear OA(41, 2, F4, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(21) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- OOA 2-folding [i] based on linear OA(4115, 16394, F4, 22) (dual of [16394, 16279, 23]-code), using
(93, 93+22, 3222220)-Net in Base 4 — Upper bound on s
There is no (93, 115, 3222221)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1725 438538 478564 019479 408201 621072 106806 141719 309584 159939 306316 112764 > 4115 [i]