Best Known (87−16, 87, s)-Nets in Base 4
(87−16, 87, 2050)-Net over F4 — Constructive and digital
Digital (71, 87, 2050)-net over F4, using
- net defined by OOA [i] based on linear OOA(487, 2050, F4, 16, 16) (dual of [(2050, 16), 32713, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(487, 16400, F4, 16) (dual of [16400, 16313, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(487, 16401, F4, 16) (dual of [16401, 16314, 17]-code), using
- construction XX applied to Ce(16) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(41, 16, F4, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(16) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(487, 16401, F4, 16) (dual of [16401, 16314, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(487, 16400, F4, 16) (dual of [16400, 16313, 17]-code), using
(87−16, 87, 10052)-Net over F4 — Digital
Digital (71, 87, 10052)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(487, 10052, F4, 16) (dual of [10052, 9965, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(487, 16401, F4, 16) (dual of [16401, 16314, 17]-code), using
- construction XX applied to Ce(16) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- 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(41, 16, F4, 1) (dual of [16, 15, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(16) ⊂ Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(487, 16401, F4, 16) (dual of [16401, 16314, 17]-code), using
(87−16, 87, 4425584)-Net in Base 4 — Upper bound on s
There is no (71, 87, 4425585)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 23945 275991 760676 704649 766094 708342 403250 058342 561562 > 487 [i]