Best Known (59, 59+14, s)-Nets in Base 4
(59, 59+14, 2342)-Net over F4 — Constructive and digital
Digital (59, 73, 2342)-net over F4, using
- net defined by OOA [i] based on linear OOA(473, 2342, F4, 14, 14) (dual of [(2342, 14), 32715, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(473, 16394, F4, 14) (dual of [16394, 16321, 15]-code), using
- construction XX applied to Ce(13) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- linear OA(471, 16384, F4, 14) (dual of [16384, 16313, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(457, 16384, F4, 11) (dual of [16384, 16327, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(13) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- OA 7-folding and stacking [i] based on linear OA(473, 16394, F4, 14) (dual of [16394, 16321, 15]-code), using
(59, 59+14, 8197)-Net over F4 — Digital
Digital (59, 73, 8197)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(473, 8197, F4, 2, 14) (dual of [(8197, 2), 16321, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(473, 16394, F4, 14) (dual of [16394, 16321, 15]-code), using
- construction XX applied to Ce(13) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- linear OA(471, 16384, F4, 14) (dual of [16384, 16313, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(457, 16384, F4, 11) (dual of [16384, 16327, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(13) ⊂ Ce(12) ⊂ Ce(10) [i] based on
- OOA 2-folding [i] based on linear OA(473, 16394, F4, 14) (dual of [16394, 16321, 15]-code), using
(59, 59+14, 2140040)-Net in Base 4 — Upper bound on s
There is no (59, 73, 2140041)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 89 203209 405597 240380 611233 142617 634838 065768 > 473 [i]