Chapter Summary

Chapter Summary

Key Points

• 1.

  Finite-field interference alignment delivers the DoF gain without the wireless baggage. Each of $K$ users achieves $1/2$ DoF in the large-signal-dimension limit — a factor of $K/2$ over TDMA — by jointly choosing precoders so that unwanted interferers at each receiver share a common subspace. The discrete setting avoids the CSI-precision barrier that limits wireless IA.

• 2.

  Coded matrix multiplication has recovery threshold $K = pq$. When $\\mathbf{A}$ is partitioned into $p$ blocks and $\\mathbf{B}$ into $q$ blocks, the product has $pq$ blocks and the master needs at least $pq$ worker responses. Both generic random IA and deterministic polynomial codes (Chapter 5) achieve this bound.

• 3.

  Coded caching delivers a global gain $1 + KM/F$ via XOR alignment. The Maddah-Ali / Niesen scheme encodes $1 + KM/F$ user demands into a single broadcast; each user's cache cancels the unintended interferers. The same algebraic pattern will reappear as the coded-shuffling gain in Chapter 7.

• 4.

  Classical PIR capacity is an IA-tight expression. Sun-Jafar's $C_{\\text{PIR}} = (1 + 1/N + \\cdots + 1/N^{F-1})^{-1}$ is achieved by finite-field alignment of the queries across $N$ replicated databases, and matched by a cut-set converse. Chapter 13 extends this to coded storage, colluding databases, and symmetric PIR.

• 5.

  IA is the tool; the rest of the book is the specializations. Chapters 5, 7, and 13 each take the IA machinery of this chapter and specialize it to one operational setting. Understanding §4.1's alignment condition now pays dividends for most of the rest of the book.