Coding and Algorithms for Memories Lecture 5 1

Write-Once Memories (WOM) Introduced by Rivest and Shamir, “How to reuse a write-once memory”, 1982 The memory elements represent bits (2 levels) and are irreversibly programmed from ‘0’ to ‘1’ Q: How many cells are required to write 100 bits twice? P1: Is it possible to do better…? P2: How many cells to write k bits twice? P3: How many cells to write k bits t times? P3’: What is the total number of bits that is possible to write in n cells in t writes? 1 st Write 2 nd Write 2

Definition: WOM Codes Definition: An [n,t;M 1,…,M t ] t-write WOM code is a coding scheme which consists of n cells and guarantees any t writes of alphabet size M 1,…,M t by programming cells from zero to one – A WOM code consists of t encoding and decoding maps E i, D i, 1 ≤i≤ t –E 1 : {1,…,M 1 }  {0,1} n – For 2 ≤i≤ t, E i : {1,…,M i }×{0,1} n  {0,1} n such that for all (m,c) ∊ {1,…,M i }×{0,1} n, E i (m,c) ≥ c – For 1 ≤i≤ t, D i : {0,1} n  {1,…,M i } such that for D i ( E i (m,c)) =m for all (m,c) ∊ {1,…,M i }×{0,1} n The sum-rate of the WOM code is R = (Σ 1 t logM i )/n Rivest Shamir: [3,2;4,4], R = (log4+log4)/3=4/3 3

The Capacity of WOM Codes The Capacity Region for two writes C 2-WOM ={(R 1,R 2 )| ∃ p ∊ [0,0.5],R 1 ≤h(p), R 2 ≤1-p} h(p) – the entropy function h(p) = -plog(p)-(1-p)log(1-p) – p – the prob to program a cell on the 1 st write, thus R 1 ≤ h(p) – After the first write, 1-p out of the cells aren’t programmed, thus R 2 ≤ 1-p The maximum achievable sum-rate is max p ∊ [0,0.5] {h(p)+(1-p)} = log3 achieved for p=1/3: R 1 = h(1/3) = log(3)-2/3 R 2 = 1-1/3 = 2/3 4

The Capacity of WOM Codes The Capacity Region for two writes C 2-WOM ={(R 1,R 2 )| ∃ p ∊ [0,0.5],R 1 ≤h(p), R 2 ≤1-p} h(p) – the entropy function h(p) = -plog(p)-(1-p)log(1-p) The Capacity Region for t writes: C t-WOM ={(R 1,…,R t )| ∃ p 1,p 2,…p t-1 ∊ [0,0.5], R 1 ≤ h(p 1 ), R 2 ≤ (1–p 1 )h(p 2 ),…, R t-1 ≤ (1–p 1 )  (1–p t–2 )h(p t–1 ) R t ≤ (1–p 1 )  (1–p t–2 )(1–p t–1 )} p 1 - prob to prog. a cell on the 1 st write: R 1 ≤ h(p) p 2 - prob to prog. a cell on the 2 nd write (from the remainder): R 2 ≤(1-p 1 )h(p 2 ) p t-1 - prob to prog. a cell on the (t-1) th write (from the remainder): R t-1 ≤ (1–p 1 )  (1–p t–2 )h(p t–1 ) R t ≤ (1–p 1 )  (1–p t–2 )(1–p t–1 ) because (1–p 1 )  (1–p t–2 )(1–p t–1 ) cells weren’t programmed The maximum achievable sum-rate is log(t+1) 5

James Saxe’s WOM Code [n,n/2-1; n/2,n/2-1,n/2-2,…,2] WOM Code – Partition the memory into two parts of n/2 cells each – First write: input symbol m ∊ {1,…,n/2} program the i th cell of the 1 st group – The i th write, i≥2: input symbol m ∊ {1,…,n/2-i+1} copy the first group to the second group program the i th available cell in the 1 st group – Decoding: There is always one cell that is programmed in the 1 st and not in the 2 nd group Its location, among the non-programmed cells, is the message value – Sum-rate: (log(n/2)+log(n/2-1)+ … +log2)/n=log((n/2)!)/n ≈ (n/2log(n/2))/n ≈ (log n)/2 6

The Coset Coding Scheme Cohen, Godlewski, and Merkx ‘86 – The coset coding scheme – Use Error Correcting Codes (ECC) in order to construct WOM-codes – Let C[n,n-r] be an ECC with parity check matrix H of size r×n – Write r bits: Given a syndrome s of r bits, find a length-n vector e such that H ⋅ e T = s – Use ECC’s that guarantee on successive writes to find vectors that do not overlap with the previously programmed cells – The goal is to find a vector e of minimum weight such that only 0s flip to 1s 7

The Coset Coding Scheme C[n,n-r] is an ECC with an r×n parity check matrix H Write r bits: Given a syndrome s of r bits, find a length-n vector e such that H ⋅ e T = s Example: H is the parity check matrix of a Hamming code – s=100, v 1 = : c = – s=000, v 2 = : c = – s=111, v 3 = : c = – s=010, …  can’t write! This matrix gives a [7,3:8,8,8] WOM code 8

Binary Two-Write WOM-Codes First Write: program only vectors v such that rank(H v ) = r V C = { v ∊ {0,1} n | rank(H v ) = r} – For H we get |V C | = 92 - we can write 92 messages – Assume we write v 1 = v 1 = ( ) 9

Binary Two-Write WOM-Codes First Write: program only vectors v such that rank(H v ) = r, V C = { v ∊ {0,1} n | rank(H v ) = r} Second Write Encoding: Second Write Decoding: Multiply the received word by H: H ⋅ (v 1 + v 2 ) = H ⋅ v 1 + H ⋅ v 2 = s 1 + (s 1 + s 2 ) = s 2 v 1 = ( ) 1.Write a vector s 2 of r bits 2.Calculate s 1 = H ⋅ v 1 3.Find v 2 such that H v 1 ⋅ v 2 = s 1 +s 2 a 4. v 2 exists since rank(H v 1 ) = r a 5.Write v 1 +v 2 to memory 1. s 2 = s 1 = H ⋅ v 1 = H v 1 ⋅ v 2 = s 1 +s 2 = 011 a 4. v 2 = v 1 +v 2 =

Sum-rate Results The construction works for any linear code C For any C[n,n-r] with parity check matrix H, V C = { v ∊ {0,1} n | rank(H v ) = r} The rate of the first write is: R 1 (C) = (log 2 |V C |)/n The rate of the second write is: R 2 (C) = r/n Thus, the sum-rate is: R(C) = (log 2 |V C | + r)/n In the last example: – R 1 = log(92)/7=6.52/7=0.93, R 2 =3/7=0.42, R=1.35 Goal: Choose a code C with parity check matrix H that maximizes the sum-rate 11

Capacity Achieving Results The Capacity region C 2-WOM ={(R 1,R 2 )| ∃ p ∊ [0,0.5],R 1 ≤h(p), R 2 ≤1-p} Theorem: For any (R 1, R 2 ) ∊ C 2-WOM and ε>0, there exists a linear code C satisfying R 1 (C) ≥ R 1 -ε and R 2 (C) ≥ R 2 –ε By computer search – Best unrestricted sum-rate (upper bound 1.58) – Best fixed sum-rate (upper bound 1.54) 12

Capacity Region and Achievable Rates of Two-Write WOM codes 13

Non-Binary WOM Codes Definition: An [n,t; M 1,…,M t ] q t-write WOM code is a coding scheme that consists of n q-ary cells and guarantees any t writes of alphabet size M 1,…,M t only by increasing the cell levels The sum-rate of the WOM-code is R = (Σ i=1 t logM i )/n 14

WOM Capacity The capacity of non-binary WOM-codes was given by Fu and Han Vinck, ‘99 The maximal sum-rate using t writes and q-ary cells is There is no tight upper bound on the sum-rate in case equal amount of information is written on each write 15

Basic Constructions Construction 1: Assume t=q–1 and there are n q-ary cells – On each write, n binary bits are written – Write to the cells level by level – The sum-rate is n(q–1)/n = q-1 (upper bound ≈ 2(q-1)) Construction 2: Assume t=2 and q is odd – First write: use only levels 0,1,…,(q-1)/2 – Second write: use only levels (q-1)/2,…,q-1 – The sum-rate is 2log((q+1)/2) – The difference between the upper bound is 16

Construction 3 Assume q is a power of 2, q=2 m Every cell is converted into m bits For 1 ≤ i ≤ m, the i th bits from each cell comprise a t-write binary WOM code The m WOM codes write into their corresponding bits in the cells independently Since every bit can only change from 0 to 1, the level of each cell cannot decrease – Use the binary expansion ϕ :{0,1} m → {0,…,2 m -1} x=(x 0,…,x m-1 ), ϕ (x) = ∑ j 2 m-j x j 17

Example – Construction 3 Let q = 8 = 2 3 and n = 3 Every cell corresponds to 3 bits Use Rivest-Shamir WOM-code The i th bits from each cell comprise a WOM-code The sum-rate is 6∙2/3 = 4 (upper bound is 5.17) Write NumberData bits Encoding by the RS base-code Encoded values in the 8-ary cells 1(11,01,10)(100,001,010)(100,001,010)(4,1,2)(4,1,2) 2(00,11,01)(101,111,110)(101,111,110)(7,3,6)(7,3,6) 18

Construction 3 Theorem: If q=2 m and there exists a t-write binary WOM-code of sum-rate R, then there exists a t-write q-ary WOM-code of sum-rate mR 19

Binary Two-Write WOM-Codes First Write: program only vectors v such that rank(H v ) = r, V C = { v ∊ {0,1} n | rank(H v ) = r} Second Write Encoding: Second Write Decoding: Multiply the received word by H: H ⋅ (v 1 + v 2 ) = H ⋅ v 1 + H ⋅ v 2 = s 1 + (s 1 + s 2 ) = s 2 v 1 = ( ) 1.Write a vector s 2 of r bits 2.Calculate s 1 = H ⋅ v 1 3.Find v 2 such that H v 1 ⋅ v 2 = s 1 +s 2 a 4. v 2 exists since rank(H v 1 ) = r a 5.Write v 1 +v 2 to memory 1. s 2 = s 1 = H ⋅ v 1 = H v 1 ⋅ v 2 = s 1 +s 2 = 011 a 4. v 2 = v 1 +v 2 =

Non-Binary Two-Write WOM-Codes Let C[n,n-r] be a linear code w/ p.c.m H of size r×n over GF(q) For a vector v ∊ GF(q) n, let H v be the matrix H with 0’s in the columns that correspond to the positions of the nonzeros’s in v First Write: program only vectors v such that rank(H v )=r, V C = { v ∊ GF(q) n | rank(H v ) = r} Second Write Encoding: Write a vector s 2 of r symbols over GF(q) – Let v 1 be the programmed vector on the first write, and let s 1 = H ⋅ v 1 – Remember that rank(H v 1 ) = r a – Find v 2 such that H v 1 ⋅ v 2 = - s 1 +s 2 F – A solution v 2 exists since rank(H v 1 ) = r a – Write v 1 +v 2 to memory Second Write Decoding: Multiply the received word by H: H ⋅ (v 1 + v 2 ) = H ⋅ v 1 + H ⋅ v 2 = s 1 + ( - s 1 + s 2 ) = s 2 This construction works for any linear code over GF(q) 21

Code Rate The construction works for any linear code C over GF(q) For any C[n,n-r] with parity check matrix H, V C = { v ∊ GF(q) n | rank(H v ) = r} The rate of the first write is: R 1 (C) = (log 2 |V C |)/n The rate of the second write is: R 2 (C) = r ⋅ log 2 q/n Thus, the total rate is: R(C) = (log 2 |V C | + r ⋅ log 2 q)/n By choosing the parity check matrix H uniformly at random, it is possible to achieve the following capacity region: {(R 1,R 2 )| ∃ p ∊ [0,1-1/q], R 1 ≤h(p)+p ⋅ log 2 (q-1), R 2 ≤(1-p) ⋅ log 2 q} – The maximum achievable rate log 2 (2q-1) is achieved for p=(q-1)/(2q-1) Remarks: – This is not an optimal non-binary two-write WOM code construction – Cells cannot be programmed twice – However, these codes are useful to construct multiple-write WOM codes 22

Three-Write WOM Codes Let C 3 be an [n,2; 2 nR 1,2 nR 2 ] two-write WOM code over GF(3) Construct a binary [2n,3; 2 nR 1,2 nR 2,2 n ] 3-write WOM code as follows: – The code has 2n cells, which are broken into n 2-cell blocks – First write: use the first write of the code C 3 Write a message m 1 from {1,…, 2 nR 1 } This message is supposed to be written as a ternary vector v 1 ∊ GF(3) n Write the vector v 1 into the n 2-cell blocks, using the mapping Φ :GF(3) → (GF(2),GF(2)), where Φ(0)=(0,0), Φ(1)=(1,0), Φ(2)=(0,1) – Second write: use the second write of the code C 3 Write a message m 2 from {1,…, 2 nR 2 } The ternary vector v 2 is written into the n 2-cell blocks using the mapping Φ Each 2-cell block is written at most once and at most one cell is programmed C 3 Encoder m 1 ∊ {1,…, 2 nR 1 } v 1 ∊ GF(3) n Φ u 1 ∊ {0,1} 2n C 3 Encoder m 2 ∊ {1,…, 2 nR 2 } v 2 ∊ GF(3) n Φ u 2 ∊ {0,1} 2n 23

Three-Write WOM Codes Let C 3 be an [n,2; 2 nR 1,2 nR 2 ] two-write WOM code over GF(3) Construct a binary [2n,3; 2 nR 1,2 nR 2,2 n ] 3-write WOM code as follows: – The code has 2n cells, which are broken into n 2-cell blocks – First write: use the first write of the code C 3 Write a message m 1 from {1,…, 2 nR 1 } This message is supposed to be written as a ternary vector v 1 ∊ GF(3) n Write the vector v 1 into the n 2-cell blocks, using the mapping Φ :GF(3) → (GF(2),GF(2)), where Φ(0)=(0,0), Φ(1)=(1,0), Φ(2)=(0,1) – Second write: use the second write of the code C 3 Write a message m 2 from {1,…, 2 nR 2 } The ternary vector v 2 is written into the n 2-cell blocks using the mapping Φ Each 2-cell block is written at most once and at most one cell is programmed – Third Write: write n bits Each bit is stored in a 2-cell block: the bit value is 1 iff both cells are programmed It is possible to write a bit in each block since at most one cell in each block was programmed 24

Three-Write WOM Code - Example Ex: C 3 is a 2-write WOM code over GF(3) with n=7 cells First Write – Write a vector v 1 over GF(3) according to the first write of C 3 – Assume the vector is v 1 = – Using the mapping Φ, v 1 is written into the 2n = 14 binary cells → Φ Second Write – Write a vector v 2 over GF(3) according to the second write of C 3 – Assume the vector is v 2 = – Using the mapping Φ, v 2 is written into the 2n = 14 binary cells → Φ Third Write: Write 7 bits – Assume the message is v 3 = , then write to memory

Code Rate Theorem: Let C 3 be an [n,2; 2 nR 1,2 nR 2 ] 2-write WOM code over GF(3) of sum-rate R 1 +R 2, then there exists a binary [2n,3; 2 nR 1,2 nR 2,2 n ] 3-write WOM code of sum- rate (R 1 +R 2 +1)/2 The best sum-rate for the code C 3 is log5 = 2.32 Conclusion: It is possible to find a binary 3-write WOM code of sum-rate (log5 +1)/2 = 1.66 Using a computer search, the best sum-rate found for WOM codes over GF(3) is 2.22 and thus there exists a 3-write WOM code of sum-rate

Four-Write WOM Codes Let C 3 be an [n,2; 2 nR 3,1,2 nR 3,2 ] two-write WOM code over GF(3) and let C 2 be an [n,2; 2 nR 2,1,2 nR 2,2 ] binary two-write WOM code Construct a [2n,4; 2 nR 1,2 nR 2, 2 nR 2,1,2 nR 2,2 ] 4-write WOM code as follows: – The code has 2n cells, which are broken into n 2-cell blocks – First write: use the first write of the code C 3 Write a message from {1,…,2 nR 3,1 } This message is supposed to be written as a ternary vector v ∊ GF(3) n Write the vector v into the n 2-cell blocks, using the mapping Φ :GF(3) → (GF(2),GF(2)), where Φ(0)=(0,0), Φ(1)=(1,0), Φ(2)=(0,1) – Second write: use the second write of the code C 3 Write a message from {1,…,2 nR 3,2 } The ternary vector is written into the n 2-cell blocks using the mapping Φ Note that each 2-cell block is written only once – Third and fourth writes: use the first and second writes of C 2 Each bit is stored in a 2-cell block: the bit value is 1 iff both cells are programmed It is possible to write a bit in each block since at most one cell in each block was programmed 27

Code Rate Theorem: Let C 3 be an [n,2; 2 nR 3,1,2 nR 3,2 ] 2-write WOM code over GF(3) (found before) of sum-rate R 1 +R 2, and let C 2 be an [n,2; 2 nR 2,1,2 nR 2,2 ] binary two-write WOM code then there exists a binary [2n,4; 2 nR 3,1,2 nR 3,2,2 nR 2,1,2 nR 2,2 ] 4-write WOM code of sum-rate (R 3,1 +R 3,2 +R 2,1 +R 2,2 )/2 a The best sum-rate for the code C 3 is log5 = 2.32 and for the code C 2 is log3 = 1.58 Conclusion: It is possible to find a binary 3-write WOM code of sum-rate (log5 + log3)/2 = 1.95 Using a computer search, the best sum-rate found for codes over GF(3) is 2.22 and for codes over GF(2) is Thus there exists a 4-write WOM code of sum-rate

Five-Write WOM-Codes Use a 2-write WOM code C 3 over GF(3) of sum-rates R 3,1, R 3,2 and a binary 3-write WOM code C 2 of sum-rates R 2,1, R 2,2, R 2,3 a The first and second writes are implemented as before using the first and second writes of the code C 3 The third, fourth, and fifth writes are also implemented as before by using the first, second, and third writes of C 2 It is possible to achieve a five-write WOM code of rate (log5+(log5+1)/2)/2=1.99 and there exists a code of rate For six writes, we will change the code C 2 to be a 4-write WOM code such that it is possible to achieve WOM-rate (log5+1.95)/2 = 2.14 and there exists a code of rate