MinT - Best Known (89, 89+11, s)-Nets in Base 8

Best Known (89, 89+11, s)-Nets in Base 8

(89, 89+11, 3360057)-Net over F₈ — Constructive and digital

Digital (89, 100, 3360057)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (13, 18, 4617)-net over F₈, using
  - net defined by OOA [i] based on linear OOA(8¹⁸, 4617, F₈, 6, 5) (dual of [(4617, 6), 27684, 6]-NRT-code), using
    - OOA stacking with additional row [i] based on linear OOA(8¹⁸, 4618, F₈, 2, 5) (dual of [(4618, 2), 9218, 6]-NRT-code), using
      - (u,Â u+v)-construction [i] based on
        linear OOA(8⁴, 585, F₈, 2, 2) (dual of [(585, 2), 1166, 3]-NRT-code), using
        appending kth column [i] based on linear OA(8⁴, 585, F₈, 2) (dual of [585, 581, 3]-code), using
        Hamming code H(4,8) [i]
        linear OOA(8¹⁴, 4033, F₈, 2, 5) (dual of [(4033, 2), 8052, 6]-NRT-code), using
        OOA 2-folding [i] based on linear OA(8¹⁴, 8066, F₈, 5) (dual of [8066, 8052, 6]-code), using
        trace code [i] based on linear OA(64⁷, 4033, F₆₄, 5) (dual of [4033, 4026, 6]-code), using
        a code by Danev and Olsson [i]
2. digital (71, 82, 3355440)-net over F₈, using
  - trace code for nets [i] based on digital (30, 41, 1677720)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64⁴¹, 1677720, F₆₄, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
      - OOA 5-folding and stacking with additional row [i] based on linear OA(64⁴¹, 8388601, F₆₄, 11) (dual of [8388601, 8388560, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(64⁴¹, large, F₆₄, 11) (dual of [large, large−41, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 16777217Â | 64⁸âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(89, 89+11, 3388080)-Net in Base 8 — Constructive

(89, 100, 3388080)-net in base 8, using

base change [i] based on digital (64, 75, 3388080)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (8, 13, 32640)-net over F₁₆, using
    - net defined by OOA [i] based on linear OOA(16¹³, 32640, F₁₆, 5, 5) (dual of [(32640, 5), 163187, 6]-NRT-code), using
      - OOA 2-folding and stacking with additional row [i] based on linear OA(16¹³, 65281, F₁₆, 5) (dual of [65281, 65268, 6]-code), using
        base reduction for projective spaces (embedding PG(6,256) in PG(12,16)) [i] based on linear OA(256⁷, 65281, F₂₅₆, 5) (dual of [65281, 65274, 6]-code), using
        a code by Danev and Olsson [i]
  2. digital (51, 62, 3355440)-net over F₁₆, using
    - trace code for nets [i] based on digital (20, 31, 1677720)-net over F₂₅₆, using
      - net defined by OOA [i] based on linear OOA(256³¹, 1677720, F₂₅₆, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
        OOA 5-folding and stacking with additional row [i] based on linear OA(256³¹, 8388601, F₂₅₆, 11) (dual of [8388601, 8388570, 12]-code), using
        discarding factors / shortening the dual code based on linear OA(256³¹, large, F₂₅₆, 11) (dual of [large, large−31, 12]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,10], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 12 [i]

(89, 89+11, large)-Net over F₈ — Digital

Digital (89, 100, large)-net over F₈, using

8³ times duplication [i] based on digital (86, 97, large)-net over F₈, using
- t-expansion [i] based on digital (83, 97, large)-net over F₈, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8⁹⁷, large, F₈, 14) (dual of [large, large−97, 15]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 8⁸âˆ’1, defining interval IÂ =Â [0,13], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 15 [i]

(89, 89+11, large)-Net in Base 8 — Upper bound on s

There is no (89, 100, large)-net in base 8, because

9 times m-reduction [i] would yield (89, 91, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]

Best Known (89, 89+11, s)-Nets in Base 8

(89, 89+11, 3360057)-Net over F8 — Constructive and digital

(89, 89+11, 3388080)-Net in Base 8 — Constructive

(89, 89+11, large)-Net over F8 — Digital

(89, 89+11, large)-Net in Base 8 — Upper bound on s

(89, 89+11, 3360057)-Net over F₈ — Constructive and digital

(89, 89+11, large)-Net over F₈ — Digital