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John Kubiatowicz Electrical Engineering and Computer Sciences University of California, Berkeley

CS252 Graduate Computer Architecture Lecture 23 Memory Technology (Con’t) Error Correction Codes April 21 st , 2010. John Kubiatowicz Electrical Engineering and Computer Sciences University of California, Berkeley http://www.eecs.berkeley.edu/~kubitron/cs252. Main Memory Background.

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John Kubiatowicz Electrical Engineering and Computer Sciences University of California, Berkeley

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  1. CS252Graduate Computer ArchitectureLecture 23Memory Technology (Con’t)Error Correction CodesApril 21st, 2010 John Kubiatowicz Electrical Engineering and Computer Sciences University of California, Berkeley http://www.eecs.berkeley.edu/~kubitron/cs252

  2. Main Memory Background • Performance of Main Memory: • Latency: Cache Miss Penalty • Access Time: time between request and word arrives • Cycle Time: time between requests • Bandwidth: I/O & Large Block Miss Penalty (L2) • Main Memory is DRAM: Dynamic Random Access Memory • Dynamic since needs to be refreshed periodically (8 ms, 1% time) • Addresses divided into 2 halves (Memory as a 2D matrix): • RAS or Row Address Strobe • CAS or Column Address Strobe • Cache uses SRAM: Static Random Access Memory • No refresh (6 transistors/bit vs. 1 transistorSize: DRAM/SRAM ­ 4-8, Cost/Cycle time: SRAM/DRAM ­ 8-16 cs252-S10, Lecture 23

  3. DRAM Architecture bit lines word lines Col. 1 Col.2M Row 1 N Row Address Decoder Row 2N Memory cell(one bit) M N+M Column Decoder & Sense Amplifiers D Data • Bits stored in 2-dimensional arrays on chip • Modern chips have around 4 logical banks on each chip • each logical bank physically implemented as many smaller arrays cs252-S10, Lecture 23

  4. 1-T Memory Cell (DRAM) • Write: • 1. Drive bit line • 2.. Select row • Read: • 1. Precharge bit line to Vdd/2 • 2.. Select row • 3. Cell and bit line share charges • Very small voltage changes on the bit line • 4. Sense (fancy sense amp) • Can detect changes of ~1 million electrons • 5. Write: restore the value • Refresh • 1. Just do a dummy read to every cell. row select bit cs252-S10, Lecture 23

  5. DRAM Capacitors: more capacitance in a small area • Trench capacitors: • Logic ABOVE capacitor • Gain in surface area of capacitor • Better Scaling properties • Better Planarization • Stacked capacitors • Logic BELOW capacitor • Gain in surface area of capacitor • 2-dim cross-section quite small cs252-S10, Lecture 23

  6. DRAM Operation: Three Steps • Precharge • charges bit lines to known value, required before next row access • Row access (RAS) • decode row address, enable addressed row (often multiple Kb in row) • bitlines share charge with storage cell • small change in voltage detected by sense amplifiers which latch whole row of bits • sense amplifiers drive bitlines full rail to recharge storage cells • Column access (CAS) • decode column address to select small number of sense amplifier latches (4, 8, 16, or 32 bits depending on DRAM package) • on read, send latched bits out to chip pins • on write, change sense amplifier latches. which then charge storage cells to required value • can perform multiple column accesses on same row without another row access (burst mode) cs252-S10, Lecture 23

  7. RAS_L CAS_L WE_L OE_L A 256K x 8 DRAM D 9 8 RAS_L DRAM Read Timing (Example) • Every DRAM access begins at: • The assertion of the RAS_L • 2 ways to read: early or late v. CAS DRAM Read Cycle Time CAS_L A Row Address Col Address Junk Row Address Col Address Junk WE_L OE_L D High Z Junk Data Out High Z Data Out Read Access Time Output Enable Delay Early Read Cycle: OE_L asserted before CAS_L Late Read Cycle: OE_L asserted after CAS_L cs252-S10, Lecture 23

  8. Main Memory Performance • DRAM (Read/Write) Cycle Time >> DRAM (Read/Write) Access Time • ­ 2:1; why? • DRAM (Read/Write) Cycle Time : • How frequent can you initiate an access? • Analogy: A little kid can only ask his father for money on Saturday • DRAM (Read/Write) Access Time: • How quickly will you get what you want once you initiate an access? • Analogy: As soon as he asks, his father will give him the money • DRAM Bandwidth Limitation analogy: • What happens if he runs out of money on Wednesday? Cycle Time Access Time Time cs252-S10, Lecture 23

  9. Increasing Bandwidth - Interleaving Access Pattern without Interleaving: CPU Memory D1 available Start Access for D1 Start Access for D2 Memory Bank 0 Access Pattern with 4-way Interleaving: Memory Bank 1 CPU Memory Bank 2 Memory Bank 3 Access Bank 1 Access Bank 0 Access Bank 2 Access Bank 3 We can Access Bank 0 again cs252-S10, Lecture 23

  10. Main Memory Performance • Simple: • CPU, Cache, Bus, Memory same width (32 bits) • Wide: • CPU/Mux 1 word; Mux/Cache, Bus, Memory N words (Alpha: 64 bits & 256 bits) • Interleaved: • CPU, Cache, Bus 1 word: Memory N Modules(4 Modules); example is word interleaved cs252-S10, Lecture 23

  11. Quest for DRAM Performance • Fast Page mode • Add timing signals that allow repeated accesses to row buffer without another row access time • Such a buffer comes naturally, as each array will buffer 1024 to 2048 bits for each access • Synchronous DRAM (SDRAM) • Add a clock signal to DRAM interface, so that the repeated transfers would not bear overhead to synchronize with DRAM controller • Double Data Rate (DDR SDRAM) • Transfer data on both the rising edge and falling edge of the DRAM clock signal  doubling the peak data rate • DDR2 lowers power by dropping the voltage from 2.5 to 1.8 volts + offers higher clock rates: up to 400 MHz • DDR3 drops to 1.5 volts + higher clock rates: up to 800 MHz • Improved Bandwidth, not Latency cs252-S10, Lecture 23

  12. Fast Memory Systems: DRAM specific • Multiple CAS accesses: several names (page mode) • Extended Data Out (EDO): 30% faster in page mode • Newer DRAMs to address gap; what will they cost, will they survive? • RAMBUS: startup company; reinvented DRAM interface • Each Chip a module vs. slice of memory • Short bus between CPU and chips • Does own refresh • Variable amount of data returned • 1 byte / 2 ns (500 MB/s per chip) • Synchronous DRAM: 2 banks on chip, a clock signal to DRAM, transfer synchronous to system clock (66 - 150 MHz) • DDR DRAM: Two transfers per clock (on rising and falling edge) • Intel claims FB-DIMM is the next big thing • Stands for “Fully-Buffered Dual-Inline RAM” • Same basic technology as DDR, but utilizes a serial “daisy-chain” channel between different memory components. cs252-S10, Lecture 23

  13. N cols 1st M-bit Access 2nd M-bit 3rd M-bit 4th M-bit RAS_L CAS_L A Row Address Col Address Col Address Col Address Col Address Fast Page Mode Operation Column Address • Regular DRAM Organization: • N rows x N column x M-bit • Read & Write M-bit at a time • Each M-bit access requiresa RAS / CAS cycle • Fast Page Mode DRAM • N x M “SRAM” to save a row • After a row is read into the register • Only CAS is needed to access other M-bit blocks on that row • RAS_L remains asserted while CAS_L is toggled DRAM Row Address N rows N x M “SRAM” M bits M-bit Output cs252-S10, Lecture 23

  14. Precharge CAS x RAS (New Bank) Burst READ CAS Latency SDRAM timing (Single Data Rate) • Micron 128M-bit dram (using 2Meg16bit4bank ver) • Row (12 bits), bank (2 bits), column (9 bits) cs252-S10, Lecture 23

  15. Double-Data Rate (DDR2) DRAM 200MHz Clock [ Micron, 256Mb DDR2 SDRAM datasheet ] Row Column Precharge Row’ Data 400Mb/s Data Rate cs252-S10, Lecture 23

  16. DDR vs DDR2 vs DDR3 • All about increasing the rate at the pins • Not an improvement in latency • In fact, latency can sometimes be worse • Internal banks often consumed for increased bandwidth cs252-S10, Lecture 23

  17. DRAM Packaging • DIMM (Dual Inline Memory Module) contains multiple chips arranged in “ranks” • Each rank has clock/control/address signals connected in parallel (sometimes need buffers to drive signals to all chips), and data pins work together to return wide word • e.g., a rank could implement a 64-bit data bus using 16x4-bit chips, or a 64-bit data bus using 8x8-bit chips. • A modern DIMM usually has one or two ranks (occasionally 4 if high capacity) • A rank will contain the same number of banks as each constituent chip (e.g., 4-8) ~7 Clock and control signals DRAM chip Address lines multiplexed row/column address ~12 Data bus (4b,8b,16b,32b) cs252-S10, Lecture 23

  18. Bank Bank Bank Bank Bank Bank Bank Bank Chip Chip Chip Chip Chip Chip Chip Chip 16 16 16 16 16 16 16 16 Rank Rank DRAM Channel 64-bit Data Bus Memory Controller Command/Address Bus cs252-S10, Lecture 23

  19. x 2 x 8 DRAM name based on Peak Chip Transfers / SecDIMM name based on Peak DIMM MBytes / Sec cs252-S10, Lecture 23

  20. Controller FB-DIMM FB-DIMM FB-DIMM FB-DIMM FB-DIMM FB-DIMM Memories • Uses Commodity DRAMs with special controller on actual DIMM board • Connection is in a serial form: Regular DIMM FB-DIMM cs252-S10, Lecture 23

  21. FLASH Memory • Like a normal transistor but: • Has a floating gate that can hold charge • To write: raise or lower wordline high enough to cause charges to tunnel • To read: turn on wordline as if normal transistor • presence of charge changes threshold and thus measured current • Two varieties: • NAND: denser, must be read and written in blocks • NOR: much less dense, fast to read and write Samsung 2007: 16GB, NAND Flash cs252-S10, Lecture 23

  22. Tunneling Magnetic Junction (MRAM) • Tunneling Magnetic Junction RAM (TMJ-RAM) • Speed of SRAM, density of DRAM, non-volatile (no refresh) • “Spintronics”: combination quantum spin and electronics • Same technology used in high-density disk-drives cs252-S10, Lecture 23

  23. Phase Change memory (IBM, Samsung, Intel) • Phase Change Memory (called PRAM or PCM) • Chalcogenide material can change from amorphous to crystalline state with application of heat • Two states have very different resistive properties • Similar to material used in CD-RW process • Exciting alternative to FLASH • Higher speed • May be easy to integrate with CMOS processes cs252-S10, Lecture 23

  24. Error Correction Codes (ECC) • Memory systems generate errors (accidentally flipped-bits) • DRAMs store very little charge per bit • “Soft” errors occur occasionally when cells are struck by alpha particles or other environmental upsets. • Less frequently, “hard” errors can occur when chips permanently fail. • Problem gets worse as memories get denser and larger • Where is “perfect” memory required? • servers, spacecraft/military computers, ebay, … • Memories are protected against failures with ECCs • Extra bits are added to each data-word • used to detect and/or correct faults in the memory system • in general, each possible data word value is mapped to a unique “code word”. A fault changes a valid code word to an invalid one - which can be detected. cs252-S10, Lecture 23

  25. ECC Approach: Redundancy • Approach: Redundancy • Add extra information so that we can recover from errors • Can we do better than just create complete copies? • Block Codes: Data Coded in blocks • k data bits coded into n encoded bits • Measure of overhead: Rate of Code: K/N • Often called an (n,k) code • Consider data as vectors in GF(2) [ i.e. vectors of bits ] • Code Space is set of all 2n vectors, Data space set of 2k vectors • Encoding function: C=f(d) • Decoding function: d=f(C’) • Not all possible code vectors, C, are valid! cs252-S10, Lecture 23

  26. General Idea: Code Vector Space Code Space • Not every vector in the code space is valid • Hamming Distance (d): • Minimum number of bit flips to turn one code word into another • Number of errors that we can detect: (d-1) • Number of errors that we can fix: ½(d-1) C0=f(v0) Code Distance (Hamming Distance) v0 cs252-S10, Lecture 23

  27. Some Code Types • Linear Codes:Code is generated by G and in null-space of H • (n,k) code: Data space 2k, Code space 2n • (n,k,d) code: specify distance d as well • Random code: • Need to both identify errors and correct them • Distance d  correct ½(d-1) errors • Erasure code: • Can correct errors if we know which bits/symbols are bad • Example: RAID codes, where “symbols” are blocks of disk • Distance d  correct (d-1) errors • Error detection code: • Distance d  detect (d-1) errors • Hamming Codes • d = 3  Columns nonzero, Distinct • d = 4 Columns nonzero, Distinct, Odd-weight • Binary Golay code: based on quadratic residues mod 23 • Binary code: [24, 12, 8] and [23, 12, 7]. • Often used in space-based schemes, can correct 3 errors cs252-S10, Lecture 23

  28. Hamming Bound, symbols in GF(2) • Consider an (n,k) code with distance d • How do n, k, and d relate to one another? • First question: How big are spheres? • For distance d, spheres are of radius ½ (d-1), • i.e. all error with weight ½ (d-1) or less must fit within sphere • Thus, size of sphere is at least: 1 + Num(1-bit err) + Num(2-bit err) + …+ Num( ½(d-1) – bit err)  • Hamming bound reflects bin-packing of spheres: • need 2k of these spheres within code space cs252-S10, Lecture 23

  29. G must be an nk matrix How to Generate code words? • Consider a linear code. Need a Generator Matrix. • Let vi be the data value (k bits), Ci be resulting code (n bits): • Are there 2k unique code values? • Only if the k columns of G are linearly independent! • Of course, need some way of decoding as well. • Is this linear??? Why or why not? • A code is systematic if the data is directly encoded within the code words. • Means Generator has form: • Can always turn non-systematiccode into a systematic one (row ops) • But – What is distance of code? Not Obvious! cs252-S10, Lecture 24

  30. Implicitly Defining Codes by Check Matrix • Consider a parity-check matrix H (n[n-k]) • Define valid code words Ci as those that give Si=0 (null space of H) • Size of null space? (null-rank H)=k if (n-k) linearly independent columns in H • Suppose we transmit code word C with error: • Model this as vector E which flips selected bits of C to get R (received): • Consider what happens when we multiply by H: • What is distance of code? • Code has distance d if no sum of d-1 or less columns yields 0 • I.e. No error vectors, E, of weight < d have zero syndromes • So – Code design is designing H matrix cs252-S10, Lecture 24

  31. P is (n-k)k, I is (n-k)(n-k) Result: H is (n-k)n P is (n-k)k, I is kk Result: G is nk How to relate G and H (Binary Codes) • Defining H makes it easy to understand distance of code, but hard to generate code (H defines code implicitly!) • However, let H be of following form: • Then, G can be of following form (maximal code size): • Notice: G generates values in null-space of H and has k independent columns so generates 2k unique values: cs252-S10, Lecture 24

  32. Parity code (8-bits): Note: Complexity of logic depends on number of 1s in row! c8 v7v6v5v4v3v2v1v0 C8 C7C6 C5 C4 C3 C2 C1 C0 + + s0 Simple example (Parity, d=2) cs252-S10, Lecture 24

  33. Repetition code (1-bit): Positives: simple Negatives: Expensive: only 33% of code word is data Not packed in Hamming-bound sense (only D=3). Could get much more efficient coding by encoding multiple bits at a time C0 v0 C1 C2 C0 Error C1 C2 Simple example: Repetition (voting, D=3) cs252-S10, Lecture 24

  34. Simple Example: Hamming Code (d=3) • Binary Hamming code meets Hamming bound • Recall bound for d=3: • So, rearranging: • Thus, for: • c=3 check bits, k ≤ 4 • c=4 check bits, k ≤ 11, use k=8? • c=5 check bits, k ≤ 26, use k=16? • c=6 check bits, k ≤ 57, use k=32? • c=7 check bits, k ≤ 120, use k=64? • H matrix consists of all unique, non-zero vectors • There are 2c-1 vectors, c used for parity, so remaining 2c-c-1 cs252-S10, Lecture 24

  35. Example, d=4 code (SEC-DED) • Design H with: • All columns non-zero, odd-weight, distinct • Note that odd-weight refers to Hamming Weight, i.e. number of zeros • Why does this generate d=4? • Any single bit error will generate a distinct, non-zero value • Any double error will generate a distinct, non-zero value • Why? Add together two distinct columns, get distinct result • Any triple error will generate a non-zero value • Why? Add together three odd-weight values, get an odd-weight value • So: need four errors before indistinguishable from code word • Because d=4: • Can correct 1 error (Single Error Correction, i.e. SEC) • Can detect 2 errors (Double Error Detection, i.e. DED) • Example: • Note: log size of nullspace will be (columns – rank) = 4, so: • Rank = 4, since rows independent, 4 cols indpt • Clearly, 8 bits in code word • Thus: (8,4) code cs252-S10, Lecture 24

  36. Tweeks: • No reason cannot make code shorter than required • Suppose n-k=8 bits of parity. What is max code size (n) for d=4? • Maximum number of unique, odd-weight columns: 27 = 128 • So, n = 128. But, then k = n – (n – k) = 120. Weird! • Just throw out columns of high weight and make (72, 64) code! • Circuit optimization: if throwing out column vectors, pick ones of highest weight (# bits=1) to simplify circuit • But – shortened codes like this might have d > 4 in some special directions • Example: Kaneda paper, catches failures of groups of 4 bits • Good for catching chip failures when DRAM has groups of 4 bits • What about EVENODD code? • Can be used to handle two erasures • What about two dead DRAMs? Yes, if you can really know they are dead cs252-S10, Lecture 24

  37. How to correct errors? • Consider a parity-check matrix H (n[n-k]) • Compute the following syndrome Si given code element Ci: • Suppose that two correctableerror vectors E1 and E2 produce same syndrome: • But, since both E1 and E2 have  (d-1)/2 bits set, E1 + E2  d-1 bits set so this conclusion cannot be true! • So, syndrome is unique indicator of correctable error vectors cs252-S10, Lecture 24

  38. cs252-S10, Lecture 24

  39. Galois Field • Definition: Field: a complete group of elements with: • Addition, subtraction, multiplication, division • Completely closed under these operations • Every element has an additive inverse • Every element except zero has a multiplicative inverse • Examples: • Real numbers • Binary, called GF(2)  Galois Field with base 2 • Values 0, 1. Addition/subtraction: use xor. Multiplicative inverse of 1 is 1 • Prime field, GF(p)  Galois Field with base p • Values 0 … p-1 • Addition/subtraction/multiplication: modulo p • Multiplicative Inverse: every value except 0 has inverse • Example: GF(5): 11  1 mod 5, 23  1mod 5, 44  1 mod 5 • General Galois Field: GF(pm)  base p (prime!), dimension m • Values are vectors of elements of GF(p) of dimension m • Add/subtract: vector addition/subtraction • Multiply/divide: more complex • Just like read numbers but finite! • Common for computer algorithms: GF(2m) cs252-S10, Lecture 24

  40. Consider polynomials whose coefficients come from GF(2). Each term of the form xnis either present or absent. Examples:0, 1, x, x2, and x7 + x6 + 1 = 1·x7 + 1· x6 + 0 · x5 + 0 · x4 + 0 · x3 + 0 · x2 + 0 · x1 + 1· x0 With addition and multiplication these form a “ring” (not quite a field – still missing division): “Add”: XOR each element individually with no carry: x4 + x3 + + x + 1 + x4 + + x2 + x x3 + x2 + 1 “Multiply”: multiplying by x is like shifting to the left. x2 + x + 1 x + 1 x2 + x + 1 x3 + x2 + x x3 + 1 Specific Example: Galois Fields GF(2n) cs252-S10, Lecture 24

  41. x4 + x3 So what about division (mod) x4 + x2 = x3 + x with remainder 0 x x4 + x2 + 1 = x3 + x2 with remainder 1 X + 1 x3 + x2 + 0x + 0 x4 + 0x3 + x2 + 0x + 1 X + 1 x3 + x2 x3 + x2 0x2 + 0x 0x + 1 Remainder 1 cs252-S10, Lecture 24

  42. Producing Galois Fields • These polynomials form a Galois (finite) field if we take the results of this multiplication modulo a prime polynomial p(x) • A prime polynomial cannot be written as product of two non-trivial polynomials q(x)r(x) • For any degree, there exists at least one prime polynomial. • With it we can form GF(2n) • Every Galois field has a primitive element, , such that all non-zero elements of the field can be expressed as a power of  • Certain choices of p(x) make the simple polynomial x the primitive element. These polynomials are called primitive • For example, x4 + x + 1 is primitive. So  = x is a primitive element and successive powers of will generate all non-zero elements of GF(16). • Example on next slide. cs252-S10, Lecture 24

  43. 0 = 1 1 = x 2 = x2 3 = x3 4 = x + 1 5 = x2 + x 6 = x3 + x2 7 = x3 + x + 1 8 = x2 + 1 9 = x3 + x 10 = x2 + x + 1 11 = x3 + x2 + x 12 = x3 + x2 + x + 1 13 = x3 + x2 + 1 14 = x3 + 1 15 = 1 Primitive element α = x in GF(2n) In general finding primitive polynomials is difficult. Most people just look them up in a table, such as: Galois Fields with primitive x4 + x + 1 α4 = x4 mod x4 + x + 1 = x4 xor x4 + x + 1 = x + 1 cs252-S10, Lecture 24

  44. x2 + x +1 x3 + x +1 x4 + x +1 x5 + x2 +1 x6 + x +1 x7 + x3 +1 x8 + x4 + x3 + x2 +1 x9 + x4 +1 x10 + x3 +1 x11 + x2 +1 Primitive Polynomials x12 + x6 + x4 + x +1 x13 + x4 + x3 + x +1 x14 + x10 + x6 + x +1 x15 + x +1 x16 + x12 + x3 + x +1 x17 + x3 + 1 x18 + x7 + 1 x19 + x5 + x2 + x+ 1 x20 + x3 + 1 x21 + x2 + 1 x22 + x +1 x23 + x5 +1 x24 + x7 + x2 + x +1 x25 + x3 +1 x26 + x6 + x2 + x +1 x27 + x5 + x2 + x +1 x28 + x3 + 1 x29 + x +1 x30 + x6 + x4 + x +1 x31 + x3 + 1 x32 + x7 + x6 + x2 +1 Galois Field Hardware Multiplication by x  shift left Taking the result mod p(x) XOR-ing with the coefficients of p(x) when the most significant coefficient is 1. Obtaining all 2n-1 non-zero elements by evaluating xk Shifting and XOR-ing 2n-1 times. for k = 1, …, 2n-1 cs252-S10, Lecture 24

  45. Reed-Solomon Codes • Galois field codes: code words consist of symbols • Rather than bits • Reed-Solomon codes: • Based on polynomials in GF(2k) (I.e. k-bit symbols) • Data as coefficients, code space as values of polynomial: • P(x)=a0+a1x1+… ak-1xk-1 • Coded: P(0),P(1),P(2)….,P(n-1) • Can recover polynomial as long as get any k of n • Properties: can choose number of check symbols • Reed-Solomon codes are “maximum distance separable” (MDS) • Can add d symbols for distance d+1 code • Often used in “erasure code” mode: as long as no more than n-k coded symbols erased, can recover data • Side note: Multiplication by constant in GF(2k) can be represented by kk matrix: ax • Decompose unknown vector into k bits: x=x0+2x1+…+2k-1xk-1 • Each column is result of multiplying a by 2i cs252-S10, Lecture 24

  46. Reed-Solomon Codes (con’t) • Reed-solomon codes (Non-systematic): • Data as coefficients, code space as values of polynomial: • P(x)=a0+a1x1+… a6x6 • Coded: P(0),P(1),P(2)….,P(6) • Called Vandermonde Matrix: maximum rank • Different representation(This H’ and G not related) • Clear that all combinations oftwo or less columns independent  d=3 • Very easy to pick whatever d you happen to want: add more rows • Fast, Systematic version of Reed-Solomon: • Cauchy Reed-Solomon, others cs252-S10, Lecture 24

  47. Aside: Why erasure coding?High Durability/overhead ratio! • Exploit law of large numbers for durability! • 6 month repair, FBLPY: • Replication: 0.03 • Fragmentation: 10-35 Fraction Blocks Lost Per Year (FBLPY) cs252-S10, Lecture 24

  48. Statistical Advantage of Fragments • Latency and standard deviation reduced: • Memory-less latency model • Rate ½ code with 32 total fragments cs252-S10, Lecture 24

  49. Conclusion • Main memory is Dense, Slow • Cycle time > Access time! • Techniques to optimize memory • Wider Memory • Interleaved Memory: for sequential or independent accesses • Avoiding bank conflicts: SW & HW • DRAM specific optimizations: page mode & Specialty DRAM • ECC: add redundancy to correct for errors • (n,k,d)  n code bits, k data bits, distance d • Linear codes: code vectors computed by linear transformation • Erasure code: after identifying “erasures”, can correct • Reed-Solomon codes • Based on GF(pn), often GF(2n) • Easy to get distance d+1 code with d extra symbols • Often used in erasure mode cs252-S10, Lecture 23

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