Self-Assessment Quiz: Classical Error Correction Review

Twenty questions on repetition codes, Hamming codes, linear codes, and the classical foundations that quantum codes build on. Aim for 16+.

Question 1

A $[n, k, d]$ linear code encodes:

A) $k$ bits into $n$ bits with minimum distance $d$ B) $n$ bits into $k$ bits C) $d$ bits into $n$ bits D) $n$ bits into $d$ bits

Question 2

A code with minimum distance $d$ can correct up to:

A) $d$ errors B) $\lfloor (d-1)/2 \rfloor$ errors C) $d-1$ errors D) $2d$ errors

Question 3

The same code can detect up to:

A) $d-1$ errors B) $d$ errors C) $\lfloor d/2 \rfloor$ D) 1 error

Question 4

The 3-bit repetition code is:

A) $[3,1,3]$ B) $[1,3,3]$ C) $[3,3,1]$ D) $[7,4,3]$

Question 5

The Hamming(7,4) code has parameters:

A) $[7,4,3]$ B) $[7,3,4]$ C) $[4,7,3]$ D) $[7,4,4]$

Question 6

It can therefore correct:

A) 1 error B) 2 errors C) 3 errors D) 0 errors

Question 7

The parity-check matrix $H$ satisfies, for any codeword $c$:

A) $Hc = 0$ B) $Hc = c$ C) $Hc = 1$ D) $H = c$

Question 8

The syndrome of a received word $r = c + e$ is:

A) $Hr = He$ — depends only on the error, not the codeword B) $Hr = c$ C) $Hr = r$ D) Undefined

Question 9

That property matters because:

A) The decoder can identify the error without knowing which codeword was sent B) It reduces code length C) It increases rate D) It eliminates errors

Question 10

The code rate is:

A) $k/n$ B) $n/k$ C) $d/n$ D) $1/d$

Question 11

Shannon's channel coding theorem states that reliable communication is possible at rates:

A) Below the channel capacity B) Above capacity C) At any rate D) Only at rate 0

Question 12

The repetition code's rate as $n$ grows is:

A) $1/n \to 0$ — asymptotically useless B) Constant C) Increasing D) $1$

Question 13

LDPC codes are notable for:

A) Sparse parity-check matrices enabling efficient iterative decoding near capacity B) Being the shortest codes C) Requiring no decoder D) Correcting all errors

Question 14

A code is linear if:

A) The sum of any two codewords is a codeword B) It is short C) It has odd length D) It corrects one error

Question 15

Two classical codes $C_2 \subset C_1$ are used to construct:

A) CSS quantum codes B) Repetition codes C) Hamming codes D) Turbo codes

Question 16

True or false: A code that detects $d-1$ errors can also correct them.

Question 17

True or false: The syndrome identifies the error uniquely for any number of errors.

Question 18

True or false: Classical codes handle only one error type, while quantum codes handle two.

Question 19

Short answer. Explain why syndrome decoding works without knowing the transmitted codeword.

Question 20

Short answer. Why is the repetition code a poor classical code but a useful pedagogical starting point for quantum codes?

Answer Key

Q Ans Note
1 A The standard notation.
2 B You need distance $2t+1$ to correct $t$ errors.
3 A Detection is easier than correction: any error of weight $< d$ moves you off the codeword.
4 A 3 bits encode 1, distance 3.
5 A The classic single-error-correcting code.
6 A $\lfloor(3-1)/2\rfloor = 1$.
7 A Codewords lie in the null space of $H$.
8 A $Hr = H(c+e) = Hc + He = He$.
9 A The syndrome depends only on the error — the property that makes decoding tractable and that quantum stabilizer codes replicate exactly.
10 A Information bits per transmitted bit.
11 A Below capacity, arbitrarily reliable communication is achievable.
12 A Rate $1/n$ vanishes; good codes keep rate bounded away from zero while distance grows.
13 A Sparse checks give near-linear-time belief-propagation decoding — and the same idea drives quantum LDPC codes.
14 A Closure under addition, which is what makes the parity-check formalism work.
15 A The CSS construction (Ch. 24) builds a quantum code from a nested pair of classical codes.
16 False Detection tells you an error occurred; correction requires identifying which error, which needs roughly twice the distance.
17 False Syndromes identify errors uniquely only up to the code's correction capacity. Beyond it, distinct errors share a syndrome and the decoder chooses wrongly — this is decoding failure, and it is why distance matters.
18 True Classical channels flip bits; quantum channels apply $X$, $Z$, or both, so a quantum code must handle two independent error types on the same physical qubit.
19 — Because $H$ annihilates every codeword: $Hr = H(c+e) = He$. The syndrome therefore carries no information about $c$ and depends only on $e$. The decoder maps syndrome to most-likely error, subtracts it, and recovers the codeword. Quantum stabilizer codes are built on exactly this property — stabilizer measurements reveal the error while commuting with the logical operators, so they say nothing about the encoded state.
20 — Poor because its rate $1/n$ vanishes: protecting against more errors costs proportionally more bits, and Shannon's theorem shows far better trade-offs exist. Useful pedagogically because the quantum three-qubit code is its direct analogue, and the differences illuminate what is quantum-specific — no-cloning means the encoding is entanglement rather than duplication, measurement must be of parities rather than bits, and a second error type ($Z$) exists that has no classical counterpart.