So you think you can FEC?

Mikhail “Misha” Fludkov (fludkov) – Pexip R&D

Erlend “Earl” Graff (egraff) – Pexip R&D

8th GStreamer Conference21 October 2017

Prague, Czech Republic

Outline

• Introduction to FEC

• Designing a FEC decoder element

• FEC algorithms

• XOR-based FEC

• Reed-Solomon FEC

• DEMO!

Bird’s eye view on FEC

#6 #5 #4 #3 #2 #1 #0#9 #8 #7#10#12 #11#13#18 #16 #15 #14#17

RTP video or audio packets (VP8, H264, OPUS, etc)

FEC packets with redundant data

Bird’s eye view on FEC

#6 #5 #4 #3 #2 #1 #0#9 #8 #7#10#12 #11#13#18 #16 #15 #14#17

FEC standards zoo

• RED - Redundant Audio Data (RFC 2198)

• ULPFEC based standards:

• ULPFEC – Generic Forward Error Correction (RFC 5109)

• ULPFEC (WebRTC)

• X-ULPFECUC (Skype)

• FLEX FEC - Flexible Forward Error Correction

GStreamer element for packet recovery

#6 #5 #4 #3 #2 #1 #0 FECDec #4 #3 #2 #1 #0

GstRtpBin FECDecFECDec

Where to hook it up? Before or after jitterbuffer?

here ? or here ?

Where to hook it up?

#6 #5 #4 #3 #2 #1 #0

Lost event with seq-num #4

GstRtpBin FECDec#4

#4

“What if…” it works and recovers packets! Yaaay!

#6 #5 #3 #2 #1 #0

#4

lost packet was recoveredand pushed here

send_rtcp_src

GstRtpBinFECDec

Where to hook it up to? Flex FEC case

#1 #0

#1 #0#1 #0#1 #0#1 #0

#1 #0

recv_rtp_src_100

recv_rtp_src_200

audio packet with ssrc 100

video packet with ssrc 200

Flex FEC packet with ssrc 300

GstRtpBin

recv_rtp_src_300

Doing it the right way

#6 #5 #3 #2 #0

recv_rtp_src_100

recv_rtp_src_200

recv_rtp_src_300

get_packets_for_recovery(ssrc=300, seq_num=4)

GstRtpBin

#4

RtpStorage

PexFecDec

PexFecDec

PexFecDec

Yet another “what if…”

#6 #5 #4 #3 #2 #1 #0#9 #8 #7#10#12 #11#13#18 #16 #15 #14#17

FEC Algorithms

XOR-based FEC

Reed-Solomon FEC

XOR-based FEC

XOR-based FEC

D1

D2

D3

D4

R1

R2

XOR-based FEC

D1

D2

D3

D4

R1

R2

=

=

⊕

⊕ ⊕⊕

D1

D1

D2

D2 D3 D4

D3

D2

D2

XOR-based FEC

=

=

⊕

D1

D2

D3

D4

R1

R2 ⊕ ⊕ ⊕

D1

D1 D4

𝑥

XOR-based FEC

=

=

⊕

⊕

𝑥

𝑥 ⊕ ⊕

D1

D4

R1

R2

D1

D1 D4

𝑦

𝑥

XOR-based FEC

=

=

⊕

⊕

𝑥

𝑥 ⊕ ⊕𝑦

D1

D4

R1

R2

D1

D1 D4

XOR-based FEC

=

=

⊕

⊕

𝑥

𝑦

𝑥

𝑥 ⊕ ⊕𝑦

D1

D4

R1

R2

D1

D1 D4

R1

D4

D1

D1⇔

𝑥 = ⊕

𝑦 = ⊕ ⊕𝑥 ⊕

R1

R2

XOR-based FEC

⇔𝑦 = ⊕ ⊕

=

=

⊕

⊕

𝑥

𝑦

𝑥

𝑥 ⊕ ⊕𝑦

D1

D4

R1

R2

D1

D1 D4

D1𝑥 = ⊕ R1

R1 D4 R2

XOR-based FEC

D1

D2

D3

D4

R1

R2

=

=

⊕

⊕ ⊕⊕

D1

D1

D2

D2 D3 D4

XOR-based FEC

D1

D2

D3

D4

R1

R2

=

=

⊕

⊕ ⊕⊕

D1

D1

D2

D2 D3 D4

XOR-based FEC

D1

D2

R1

R2

=

=

⊕

⊕ ⊕⊕

D1

D1

D2

D2

𝑥

𝑥

XOR-based FEC

D1

D2

R1

R2

=

=

⊕

⊕ ⊕⊕

D1

D1

D2

D2

𝑥

𝑦

𝑥 𝑦

Reed-Solomon codes

Irving S. Reed and Gustave Solomon

• Either

• Detect up to 𝑡 errors

• Detect and correct up to Τ𝑡 2 errors

• Correct up to 𝑡 erasures

𝑘 data symbols 𝑡 parity symbols

R-Sencoder

𝑘 symbols(data and/or parity)

𝑘 original data symbols

R-Serasuredecoder

Data packets

Redundancy packets

Reed-Solomon codes

Reed-Solomon codes

55 47 56 34 61 58 41 67 61 58 4d 67 59 58 64 6c 63 32 39 74 5a 53 45 68 49 51 3d 3d …

ee c9 95 a1 9d 21 a3 b3 87 53 75 a5 aa 16 4f 4b 70 7f 4b 95 70 c6 bd e8 cb 02 bf 8c …

5d 4a d7 8b 73 0f 61 54 bf bc d7 cd aa 22 d9 44 4a f6 48 8a 14 c4 0a c3 ca f4 d5 ff …

90 09 d6 a5 f3 e3 5d d3 24 3f 24 a8 1b 96 72 09 1f c8 cb d7 41 94 23 42 6e 85 2d d6 …

85 9a b2 76 8f 41 c2 37 29 ee d3 5b fd e0 b0 d1 49 58 91 2a 45 52 47 d1 82 b2 dd 0c …

Data packets

Redundancy packets

Reed-Solomon codes

55

ee

5d

90

85

Data packets

Redundancy packets

Reed-Solomon codes

47

c9

4a

09

9a

Data packets

Redundancy packets

Reed-Solomon codes

56

95

d7

d6

b2

Data packets

Redundancy packets

55 47 56 34 61 58 41 67 61 58 4d 67 59 58 64 6c 63 32 39 74 5a 53 45 68 49 51 3d 3d …

ee c9 95 a1 9d 21 a3 b3 87 53 75 a5 aa 16 4f 4b 70 7f 4b 95 70 c6 bd e8 cb 02 bf 8c …

5d 4a d7 8b 73 0f 61 54 bf bc d7 cd aa 22 d9 44 4a f6 48 8a 14 c4 0a c3 ca f4 d5 ff …

Reed-Solomon codes

90 09 d6 a5 f3 e3 5d d3 24 3f 24 a8 1b 96 72 09 1f c8 cb d7 41 94 23 42 6e 85 2d d6 …

85 9a b2 76 8f 41 c2 37 29 ee d3 5b fd e0 b0 d1 49 58 91 2a 45 52 47 d1 82 b2 dd 0c …

Data packets

Redundancy packets

Reed-Solomon codes

Reed-Solomon codes – encoding

10

50

100D1:

D2:

D3:

R1:

R2:

Reed-Solomon codes – encoding

10

50

100

5

4

2 3

5 2

Encoding coefficientsD1:

D2:

D3:

R1:

R2:

Reed-Solomon codes – encoding

10

50

100

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2

D1:

D2:

D3:

R1:

R2:

100

1050100

Reed-Solomon codes – encoding

10

50

100

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+

50 10

D1:

D2:

D3:

R1:

R2:

Reed-Solomon codes – encoding

10

50

100

5

4

2 3

5 2

Encoding coefficients

=

=

1005 × 502 × 103×+

+ +

+

1004 × 5 × 50 2 × 10

D1:

D2:

D3:

R1:

R2:

Reed-Solomon codes – encoding

10

50

100

5

4

2 3

5 2

Encoding coefficients

=

=

502 × 103×+

+ +

+

1004 × 5 × 50 2 × 10

500

D1:

D2:

D3:

R1:

R2:

Reed-Solomon codes – encoding

10

50

100

5

4

2 3

5 2

Encoding coefficients

=

=

103×+

+ +

+

1004 × 5 × 50 2 × 10

500 100

D1:

D2:

D3:

R1:

R2:

Reed-Solomon codes – encoding

10

50

100

5

4

2 3

5 2

Encoding coefficients

=

=

+

+ +

+

1004 × 5 × 50 2 × 10

500 100 30

D1:

D2:

D3:

R1:

R2:

Reed-Solomon codes – encoding

10

50

100

5

4

2 3

5 2

Encoding coefficients

=

=

+

+ +

+

1004 × 5 × 50 2 × 10

500 100 30630

D1:

D2:

D3:

R1:

R2:

400 250 20

Reed-Solomon codes – encoding

10

50

100

5

4

2 3

5 2

Encoding coefficients

=

=

+

+ +

+500 100 30630

D1:

D2:

D3:

R1:

R2:

400 250 20

Reed-Solomon codes – encoding

10

50

100

5

4

2 3

5 2

Encoding coefficients

=

=

+

+ +

+500 100 30630

670

D1:

D2:

D3:

R1:

R2:

100

1050100

Reed-Solomon codes – decoding

10

50

100

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+

50 10

630

670

D1:

D2:

D3:

R1:

R2:

100

1050100

Reed-Solomon codes – decoding

10

50

100

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+

50 10

630

670

D1:

D2:

D3:

R1:

R2:

100

100

Reed-Solomon codes – decoding

100

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+630

670

𝑥

𝑥

𝑥

D1:

D2:

D3:

R1:

R2:

100

100

Reed-Solomon codes – decoding

100

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+630

670

𝑥

𝑥

𝑥

𝑦

𝑦

𝑦

D1:

D2:

D3:

R1:

R2:

𝑦

𝑥

𝑥

100

100

Reed-Solomon codes – decoding

100

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+630

𝑥 𝑦

𝑦

670⇔

500 + 2𝑥 + 3𝑦 = 630

400 + 5𝑥 + 2𝑦 = 670

D1:

D2:

D3:

R1:

R2:

𝑦

𝑥

𝑥

100

100

Reed-Solomon codes – decoding

100

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+630

𝑥 𝑦

𝑦

670⇔

500 + 2𝑥 + 3𝑦 = 630

400 + 5𝑥 + 2𝑦 = 670

500 + 2𝑥 + 3𝑦 = 630 − 500

400 + 5𝑥 + 2𝑦 = 670 − 400

D1:

D2:

D3:

R1:

R2:

𝑦

𝑥

𝑥

100

100

Reed-Solomon codes – decoding

100

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+630

𝑥 𝑦

𝑦

670⇔

2𝑥 + 3𝑦 = 630 − 500

5𝑥 + 2𝑦 = 670 − 400

D1:

D2:

D3:

R1:

R2:

𝑦

𝑥

𝑥

100

100

Reed-Solomon codes – decoding

100

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+630

𝑥 𝑦

𝑦

670⇔

2𝑥 + 3𝑦 = 130

5𝑥 + 2𝑦 = 270

D1:

D2:

D3:

R1:

R2:

𝑦

𝑥

𝑥

100

100

Reed-Solomon codes – decoding

100

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+630

𝑥 𝑦

𝑦

670⇔

−3𝑥 + 𝑦 = −140

2𝑥 + 3𝑦 = 130

5𝑥 + 2𝑦 = 270

D1:

D2:

D3:

R1:

R2:

𝑦

𝑥

𝑥

100

100

Reed-Solomon codes – decoding

100

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+630

𝑥 𝑦

𝑦

670⇔

𝑦 = 3𝑥 − 140

2𝑥 + 3𝑦 = 130

5𝑥 + 2𝑦 = 270

D1:

D2:

D3:

R1:

R2:

𝑦

𝑥

𝑥

100

100

Reed-Solomon codes – decoding

100

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+630

𝑥 𝑦

𝑦

670⇔

𝑦 = 3𝑥 − 140

2𝑥 + 3𝑦 = 130

5𝑥 + 2𝑦 = 270⇔

2𝑥 + 3 3𝑥 − 140 = 130

2𝑥 + 9𝑥 − 420 = 130

11𝑥 = 550

𝑥 = 50

D1:

D2:

D3:

R1:

R2:

Reed-Solomon codes – summary

• Encoding• Linear combination (matrix multiplication)

• Decoding• Solving linear equation system (matrix inversion and multiplication)

But… 10

50

100

630

670

Linear combinations in ℝ

5 × 100 + 2 × 50 + 3 × 10

∈ ℝ ∈ ℝ ∈ ℝ

∈ 0,255 ∈ 0,255 ∈ 0,255

Linear combinations… not in ℝ?

5 × 100 + 2 × 50 + 3 × 10

∈ 0,255 ∈ 0,255 ∈ 0,255

Linear combinations… not in ℝ?

5 × 100 + 2 × 50 + 3 × 10

∈ 0,255

Fields

• Set of elements

• + and ⋅

• Given 𝑎 ∈ Ϝ and 𝑏 ∈ Ϝ, then…

• 𝑎 + 𝑏 ∈ Ϝ

• 𝑎 ⋅ 𝑏 ∈ Ϝ

• ℝ and ℚ – infinite fields

• Finite fields

Fields

ൗℤ 7ℤ

0,1,2,3,4,5,6

+ is addition modulo 7

⋅ is multiplication modulo 7

Fields

Multiplication table for Τℤ 7ℤ

1 2 3 4 5 6

1 1 2 3 4 5 6

2 4 6 1 3 5

3 2 5 1 4

4 2 6 3

5 4 2

6 1

Fields

Multiplication table for Τℤ 7ℤ

1 2 3 4 5 6

1 1 2 3 4 5 6

2 4 6 1 3 5

3 2 5 1 4

4 2 6 3

5 4 2

6 1

2 ⋅ 4 ≡ 8 ≡ 1 mod 7

Fields

Multiplication table for Τℤ 7ℤ

1 2 3 4 5 6

1 1 2 3 4 5 6

2 4 6 1 3 5

3 2 5 1 4

4 2 6 3

5 4 2

6 1

4 ⋅ 5 ≡ 20 ≡ 6 mod 7

Fields

• Finite fields – 𝑝𝑞 elements

• Solving linear equation systems works!

• Reed-Solomon: typically 2𝑞 elements

• 𝐺𝐹 28 – 0,255

Linear MDS codes over Τℤ 7ℤ – encoding

3

2

1D1:

D2:

D3:

R1:

R2:

Linear MDS codes over Τℤ 7ℤ – encoding

3

2

1

5

4

2 3

5 2

Encoding coefficientsD1:

D2:

D3:

R1:

R2:

Linear MDS codes over Τℤ 7ℤ – encoding

3

2

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2

D1:

D2:

D3:

R1:

R2:

1

321

Linear MDS codes over Τℤ 7ℤ – encoding

3

2

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+

2 3

D1:

D2:

D3:

R1:

R2:

Linear MDS codes over Τℤ 7ℤ – encoding

3

2

1

5

4

2 3

5 2

Encoding coefficients

=

= 14 ×

15 × 22 × 33×+

+

+

+5 × 2 2 × 3

D1:

D2:

D3:

R1:

R2:

Linear MDS codes over Τℤ 7ℤ – encoding

3

2

1

5

4

2 3

5 2

Encoding coefficients

=

= 14 ×

22 × 33×+

+

+

+5 × 2 2 × 3

D1:

D2:

D3:

R1:

R2:

5

Linear MDS codes over Τℤ 7ℤ – encoding

3

2

1

5

4

2 3

5 2

Encoding coefficients

=

= 14 ×

33×+

+

+

+5 × 2 2 × 3

D1:

D2:

D3:

R1:

R2:

5 4

Linear MDS codes over Τℤ 7ℤ – encoding

3

2

1

5

4

2 3

5 2

Encoding coefficients

=

= 14 ×

+

+

+

+5 × 2 2 × 3

D1:

D2:

D3:

R1:

R2:

5 4 2

Linear MDS codes over Τℤ 7ℤ – encoding

3

2

1

5

4

2 3

5 2

Encoding coefficients

=

= 14 ×

+

+

+

+5 × 2 2 × 3

D1:

D2:

D3:

R1:

R2:

5 4 24

Linear MDS codes over Τℤ 7ℤ – encoding

3

2

1

5

4

2 3

5 2

Encoding coefficients

=

=

+

+

+

+5 × 2 2 × 3

D1:

D2:

D3:

R1:

R2:

5 4 24

4

Linear MDS codes over Τℤ 7ℤ – encoding

3

2

1

5

4

2 3

5 2

Encoding coefficients

=

=

+

+

+

+ 2 × 3

D1:

D2:

D3:

R1:

R2:

5 4 24

4 3

Linear MDS codes over Τℤ 7ℤ – encoding

3

2

1

5

4

2 3

5 2

Encoding coefficients

=

=

+

+

+

+

D1:

D2:

D3:

R1:

R2:

5 4 24

4 3 6

Linear MDS codes over Τℤ 7ℤ – encoding

3

2

1

5

4

2 3

5 2

Encoding coefficients

=

=

+

+

+

+

D1:

D2:

D3:

R1:

R2:

5 4 24

4 3 66

1

321

Linear MDS codes over Τℤ 7ℤ – decoding

3

2

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+

2 3

4

6

D1:

D2:

D3:

R1:

R2:

1

321

Linear MDS codes over Τℤ 7ℤ – decoding

3

2

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+

2 3

4

6

D1:

D2:

D3:

R1:

R2:

1

1

Linear MDS codes over Τℤ 7ℤ – decoding

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+4

6

𝑥

𝑥

𝑥

D1:

D2:

D3:

R1:

R2:

1

1

Linear MDS codes over Τℤ 7ℤ – decoding

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+4

6

𝑥

𝑥

𝑥

𝑦

𝑦

𝑦

D1:

D2:

D3:

R1:

R2:

𝑦

𝑥

𝑥

1

1

Linear MDS codes over Τℤ 7ℤ – decoding

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+4

𝑥 𝑦

𝑦

6⇔

5 + 2𝑥 + 3𝑦 = 4

4 + 5𝑥 + 2𝑦 = 6

D1:

D2:

D3:

R1:

R2:

3 + 4 + 5𝑥 + 2𝑦 = 6 + 3

𝑦

𝑥

𝑥

1

1

Linear MDS codes over Τℤ 7ℤ – decoding

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+4

𝑥 𝑦

𝑦

6⇔

5 + 2𝑥 + 3𝑦 = 4

4 + 5𝑥 + 2𝑦 = 6

D1:

D2:

D3:

R1:

R2:

2 + 5 + 2𝑥 + 3𝑦 = 4 + 2

3 + 4 + 5𝑥 + 2𝑦 = 6 + 3

𝑦

𝑥

𝑥

1

1

Linear MDS codes over Τℤ 7ℤ – decoding

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+4

𝑥 𝑦

𝑦

6⇔

5 + 2𝑥 + 3𝑦 = 4

4 + 5𝑥 + 2𝑦 = 6

D1:

D2:

D3:

R1:

R2:

2 + 5 + 2𝑥 + 3𝑦 = 4 + 2

2𝑥 + 3𝑦 = 4 + 2

5𝑥 + 2𝑦 = 6 + 3

𝑦

𝑥

𝑥

1

1

Linear MDS codes over Τℤ 7ℤ – decoding

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+4

𝑥 𝑦

𝑦

6⇔

D1:

D2:

D3:

R1:

R2:

2𝑥 + 3𝑦 = 6

5𝑥 + 2𝑦 = 2

𝑦

𝑥

𝑥

1

1

Linear MDS codes over Τℤ 7ℤ – decoding

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+4

𝑥 𝑦

𝑦

6⇔

D1:

D2:

D3:

R1:

R2:

2𝑥 + 3𝑦 = 6

5𝑥 + 2𝑦 = 2

𝑦

𝑥

𝑥

1

1

Linear MDS codes over Τℤ 7ℤ – decoding

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+4

𝑥 𝑦

𝑦

6⇔

D1:

D2:

D3:

R1:

R2:

0𝑥 + 5𝑦 = 1

2𝑥 + 3𝑦 = 6

5𝑥 + 2𝑦 = 2

𝑦

𝑥

𝑥

1

1

Linear MDS codes over Τℤ 7ℤ – decoding

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+4

𝑥 𝑦

𝑦

6⇔

D1:

D2:

D3:

R1:

R2:

5𝑦 = 1

2𝑥 + 3𝑦 = 6

5𝑥 + 2𝑦 = 2

𝑦

𝑥

𝑥

1

1

Linear MDS codes over Τℤ 7ℤ – decoding

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+4

𝑥 𝑦

𝑦

6⇔

D1:

D2:

D3:

R1:

R2:

Multiplication table for Τℤ 7ℤ

1 2 3 4 5 6

1 1 2 3 4 5 6

2 4 6 1 3 5

3 2 5 1 4

4 2 6 3

5 4 2

6 1

5𝑦 = 1

2𝑥 + 3𝑦 = 6

5𝑥 + 2𝑦 = 2

𝑦

𝑥

𝑥

1

1

Linear MDS codes over Τℤ 7ℤ – decoding

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+4

𝑥 𝑦

𝑦

6⇔

D1:

D2:

D3:

R1:

R2:

5𝑦 = 1

Multiplication table for Τℤ 7ℤ

1 2 3 4 5 6

1 1 2 3 4 5 6

2 4 6 1 3 5

3 2 5 1 4

4 2 6 3

5 4 2

6 1

2𝑥 + 3𝑦 = 6

5𝑥 + 2𝑦 = 2

𝑦

𝑥

𝑥

1

1

Linear MDS codes over Τℤ 7ℤ – decoding

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+4

𝑥 𝑦

𝑦

6⇔

D1:

D2:

D3:

R1:

R2:

3 ⋅ 5𝑦 = 1 ⋅ 3

Multiplication table for Τℤ 7ℤ

1 2 3 4 5 6

1 1 2 3 4 5 6

2 4 6 1 3 5

3 2 5 1 4

4 2 6 3

5 4 2

6 1

2𝑥 + 3𝑦 = 6

5𝑥 + 2𝑦 = 2

𝑦

𝑥

𝑥

1

1

Linear MDS codes over Τℤ 7ℤ – decoding

1

5

4

2 3

5 2

Encoding coefficients

=

=

5 2 3

4 5 2×

× ×

× ×

×+

+ +

+4

𝑥 𝑦

𝑦

6⇔

D1:

D2:

D3:

R1:

R2:

𝑦 = 3

Multiplication table for Τℤ 7ℤ

1 2 3 4 5 6

1 1 2 3 4 5 6

2 4 6 1 3 5

3 2 5 1 4

4 2 6 3

5 4 2

6 1

FEC it up!

• GStreamer elements

• RED encoder and decoder

• ULPFEC decoder (XOR and Reed-Solomon)

• RTPStorage element

• Reed-Solomon encoder and decoder

• Only 𝐺𝐹 28

• Matrix operations

• Cauchy-RS – fast matrix inversion

• SIMD optimizations1

1 Anvin, H. P. The mathematics of RAID-6. 2009.

DEMO THYME

Benchmarks – Reed-Solomon encoding

Benchmarks – Reed-Solomon decoding

Questions?(Do you think we can FEC?)