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Quantum Engineering Year 2: Advanced Quantum Science Month 27Day 744

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Year 2·Month 27·Week 3

Day 744: Dual Containment Condition

Day 744 of 2,016~17 min read

Learning Objectives

•**Compute** the dual code C⊥ from a generator or parity check matrix
•**Verify** dual containment for specific code pairs
•**Identify** self-orthogonal and self-dual codes
•**Construct** CSS codes from weakly self-dual codes
•**Apply** MacWilliams identities to relate code distances
•**Design** CSS codes with guaranteed parameters

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Overview Schedule Learning Objectives Classical Dual Codes Definition of Dual Code Parity CheckGenerator Duality Computing the Dual Self-Orthogonal and Self-Dual Codes Definitions Hierarchy Key Theorem for CSS CSS from Self-Orthogonal Code Constructing Self-Orthogonal Codes Method 1 Direct Construction Method 2 Construction X Method 3 Puncturing and Shortening Doubly-Even Codes Definition Connection to Self-Orthogonality Famous Doubly-Even Codes Verifying Dual Containment General Case C C Practical Test Distance Bounds for CSS MacWilliams Identity CSS Distance Formula Symmetric Case Examples of Code Families Extended Hamming 8 4 4 Reed-Muller Codes Worked Examples Example 1 Verify Self-Orthogonality Example 2 Construct Self-Orthogonal Extension Example 3 CSS Parameters Practice Problems Level 1 Direct Application Level 2 Intermediate Level 3 Challenging Computational Lab Summary Key Formulas Main Takeaways Daily Checklist Preview Day 745

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