Weighted Coverings

Advanced coverage with decreasing weight sequences

Weight Configuration

Configure your decreasing weight sequence m = (m₀, m₁, ..., mₙ)

m₀:

m₁:

m₂:

m₃:

Coverage Formula

For every x ∈ Z₃ⁿ × Z₂²ⁿ⁻ⁿ₁:

Σ_i=0ⁿ mᵢ · aᵢ(x) ≥ 1

Where aᵢ(x) = number of words at Hamming distance i from x

Current weights: m = (0.5, 0.5, 0.25, 0.125)

Weighted sum = 1.375

Example: Z₄ with m₀ = m₁ = 1/2, m₂ = 1/4

The following system guarantees either:

One first prize and two third prizes, OR
Two second prizes, OR
Four third prizes

System S = {
    0000,
    1100,
    0011,
    1111
}

Coverage Analysis

Distance 0 (exact match): weight 0.5

Distance 1 (one bit different): weight 0.5

Distance 2 (two bits different): weight 0.25

Prize Distribution

🏆 1st Prize: Exact match (4 words)

🥈 2nd Prize: Distance 1 (6 words)

🥉 3rd Prize: Distance 2 (4 words)

                 Coverage Matrix
            
                            Word x
                            a₀(x)
                            a₁(x)
                            a₂(x)
                            Σ mᵢ·aᵢ(x)
                            Status

Common Presets

Information

Weighted coverings generalize simple coverings by assigning different importance to different Hamming distances.