index.md (3462B)
1 +++ 2 title = "Lecture 3" 3 template = 'page-math.html' 4 +++ 5 6 ## Generating matrices & encoding 7 8 - rank of matrix over K: num of nonzero rows in any REF 9 - dim k of code C: dim of C as subspace of Kⁿ 10 - if C has length n and dist d -- C is (n, k, d) linear code 11 - if C linear code of length n and dim k, any matrix whose rows are basis for C is generator matrix for C (must have k rows, n cols, rank k) 12 - G generator matrix ⇔ rows of G linearly independent 13 - to find a generator matrix, put codewords in matrix and reduce, then take nonzero rows 14 15 if G generator matrix for linear code C length n and dimension k, then v = u G ranges over all $2^{K}$ words in C $\forall u \text{length} k \in 2^{k}$ 16 - → C = { words u G, u in $K^{K}$ } 17 - → u₁ G = u₂ G ⇔ u₁ = u₂ 18 19 info rate of (n, k, d) code: $\frac{\log_{2} (2^{k})}{n} = \frac{k}{n}$ 20 21 ## Parity check matrices 22 H parity check matrix for linear code C if columns form basis for dual code $C^{\perp}$. 23 - if C length n dimension k, parity check matrix has n rows, n-k columns, n-k rank 24 - H parity check ⇔ columns H linearly independent 25 - if H parity check matrix for C length n, then C = {words v ∈ Kⁿ | v H = 0} 26 27 Matrix G generating and H parity check iff: 28 1. rows of G linearly independent 29 2. columns of H linearly independent 30 3. rows(G) + columns(H) = columns(G) = rows(H) 31 4. G H = 0 32 33 H parity check for C ⇔ $H^{T}$ generator for $C^{\perp}$ 34 35 $H ^{T} G ^{T} = (G H) ^{T} = 0$ 36 37  38 39 ## Equivalent codes 40 - if $G = [I_{k}, X]$, G in standard form and generator for linear code length n dimension k with standard form G, then first K digits in codeword v = u G form word u in $K^{K}$ ("information digits") 41 - you can always permute C and rearrange every word 42 - any linear code C equivalent to linear code C' having generator matrix in standard form 43 44 ## Distance of linear code 45 H parity check for linear code C. 46 47 C has distance d ⇔ any set d-1 rows of H linearly independent & at least one set d rows of H linearly dependent 48 49 ## Cosets 50 for C linear code length n, u any word length n: coset of C determined by u = C + u = {v + u | v ∈ C} 51 52 for C linear code length n, u and v words length n: 53 - u ∈ C + v → c + u = C + v 54 - u ∈ C + u 55 - u + v ∈ C → u and v in same coset 56 - u + v ∉ C → u and v in different cosets 57 - either C + u = C + v, or the two have no common words 58 - |C + u| = |C| 59 - C dimension k → exists $2^{n-k}$ different cosets of C, each with $2^{k}$ words 60 - C is one of cosets 61 62 ## MLD for linear codes 63 when word w received, choose word u of least weight in coset c + w. conclude that v = w + u sent. 64 65 for C linear code length n, H parity check for C, w and u words in Kⁿ: 66 - w H = 0 ⇔ w codeword in C 67 - w H = u H ⇔ w and u in same coset of C 68 - u error pattern in received w → u H sum rows of H corresponding to position where errors in transmission 69 - syndrome word w in Kⁿ: w H in $K^{n-k}$ 70 71 coset leader: word of least weight in coset. 72 73 standard decoding array (SDA) matches coset leader u to syndrome u H. 74 75 constructing SDA: 76 1. list all cosets for code, elect leaders 77 2. find parity check H for code 78 3. calculate syndromes u H 79 80 decoding w received: 81 1. calculate syndrome w H 82 2. find coset leader u next to syndrome w H = u H in SDA 83 3. conclude v = w + u sent 84 85 IMLD: 86 - num words closest to w = num least weight error patterns in c + w 87 - cosets with more than 1 least weight are omitted.