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Syllabus

This syllabus tracks concept coverage for the fresh-start Codex course. Heptabase's historical status is preserved in the source map, but learner progress here starts over from Lesson 01.

Status legend:

  • ready: Codex lesson content exists.
  • next: recommended starting point.
  • later: generated and ready, but not yet the next live lesson.

Topic 0 - Course Introduction and Research Orientation

Purpose: orient the learner around the Conjecture 3 barrier and explain why the five foundations belong in one course.

Topic 1 - Classical Linear Codes Over GF(2)

  • GF(2) arithmetic as algebra: next in Lesson 01
  • Vector spaces over GF(2): next in Lesson 01
  • The [7,4,3] Hamming code: next in Lesson 01
  • Generator matrices and parity-check matrices: ready in Lesson 02
  • Dual codes: ready in Lesson 02
  • Code parameters: rate, distance, and weight: ready in Lesson 02
  • Puncturing and shortening: ready in Lesson 02
  • Connection to CSS codes: ready in Lesson 03

Purpose: build the algebraic language used by CSS codes, Tanner graphs, and parity-check matrices.

Topic 2 - Tanner Graphs and Code Structure

  • From parity-check matrix to bipartite graph: ready in Lesson 03
  • Degree constraints and the LDPC condition: ready in Lesson 03
  • Girth and local tree structure: ready in Lesson 04
  • Local views and neighborhoods: ready in Lesson 04
  • Tanner codes: ready in Lesson 04
  • Small-set expansion in Tanner graphs: ready in Lesson 04
  • The Tanner graph of a CSS code: ready in Lesson 05

Purpose: translate parity-check matrices into graph objects whose expansion and routing demands can be studied.

Topic 3 - Expander Graphs and Graph Cuts

  • Regular graphs and adjacency spectra: ready in Lesson 05
  • Petersen graph as a small expander: ready in Lesson 05
  • Edge expansion and Cheeger inequality: ready in Lesson 06
  • Ramanujan graphs and optimal expanders: ready in Lesson 06
  • Bisection width: ready in Lesson 06
  • 2D grid bisection width: ready in Lesson 06
  • Planar separator theorem: ready in Lesson 06
  • Expansion mismatch: ready in Lesson 06
  • 2D grid routing tightness: ready in Lesson 06

Purpose: make precise why expander demand and 2D hardware capacity are mismatched.

Topic 4 - CSS and QLDPC Codes

  • CSS construction from classical codes: ready in Lesson 07
  • Logical operators in CSS codes: ready in Lesson 07
  • Good code parameters: ready in Lesson 07
  • Why good QLDPC codes require expander Tanner graphs: ready in Lesson 07
  • Left-right Cayley complexes: ready in Lesson 07
  • Quantum Tanner codes: ready in Lesson 07
  • Spectral expansion of the diagonal graphs: ready in Lesson 07
  • The anchor gap: from diagonal expansion to Tanner expansion: ready in Lesson 07
  • Other QLDPC families in context: ready in Lesson 07

Purpose: connect the learner's stabilizer/CSS background to modern good QLDPC constructions, especially quantum Tanner codes.

Topic 5 - Syndrome Extraction and Hardware Constraints

  • Stabilizer measurement circuits: ready in Lesson 08
  • Why hardware locality matters: ready in Lesson 08
  • SWAP gates as a routing primitive: ready in Lesson 08
  • Congestion: ready in Lesson 08
  • Dilation: ready in Lesson 08
  • The CD(T_n, G) functional: ready in Lesson 08
  • Static 2D lower bound: ready in Lesson 08
  • Expansion cut to syndrome depth: ready in Lesson 08
  • Space-depth tradeoff: ready in Lesson 08
  • What is proved versus what Conjecture 3 claims: ready in Lesson 08

Purpose: turn the graph/coding foundations into the physical syndrome-depth bottleneck.

Topic 6 - Integration: Reading the Research Map

  • Tracing the lower-bound proof chain: ready in Lesson 09
  • Locating the open gaps: ready in Lesson 09
  • Congestion-dilation versus stabilizer cut rank: ready in Lesson 09
  • Reading a research card independently: ready in Lesson 09
  • Positioning toward the next course: ready in Lesson 09

Purpose: transition from a course into independent research-map reading.